##### Class 9^{th} Mathematics Term 1 Tamilnadu Board Solution

**Exercise 3.1**- State whether the following expressions are polynomials in one variable or not.…
- Write the coefficient of x^2 and x in each of the following i. 2 + 3x - 4x^2 +…
- Write the degree of each of the following polynomials. i. 4 - 3x^2 ii. 5y + √2…
- Classify the following polynomials based on their degree. i. 3x^2 + 2x + 1 ii.…
- Give one example of a binomial of degree 27 and monomial of degree 49 and…

**Exercise 3.2**- Find the zeros of the following polynomials. i. p(x)= 4x - 1 ii. p(x) = 3x + 5…
- Find the roots of the following polynomial equations. i. x - 3 = 0 ii. 5x - 6 =…
- x^2 - 5x + 6 = 0; x = 2, 3 Verify Whether the following are roots of the…
- x^2 + 4x + 3 = 0; x = −1, 2 Verify Whether the following are roots of the…
- x^3 - 2x^2 - 5x + 6 = 0; x = 1, −2, 3 Verify Whether the following are roots of…
- x^3 - 2x^2 - x + 2 = 0; x = −1, 2, 3 Verify Whether the following are roots of…

**Exercise 3.3**- (5x^3 - 8x^2 + 5x - 7) ÷ (x - 1) Find the quotient the and remainder of the…
- (2x^2 - 3x - 14) ÷ (x + 2) Find the quotient the and remainder of the following…
- (9 + 4x + 5x^2 + 3x^3) ÷ (x+ 1) Find the quotient the and remainder of the…
- (4x^3 - 2x^2 + 6x + 7) ÷ (3 + 2x) Find the quotient the and remainder of the…
- (−18 - 9x + 7x^2) ÷ (x - 2) Find the quotient the and remainder of the…

**Exercise 3.4**- 3x^3 + 4x^2 - 5x + 8 is divided by x - 1 Find the remainder using remainder…
- 5x^3 + 2x^2 - 6x + 12 is divided by x + 2 Find the remainder using remainder…
- 2x^3 - 4x^2 + 7x + 6 is divided by x - 2 Find the remainder using remainder…
- 4x^3 - 3x^2 + 2x - 4 is divided by x + 3 Find the remainder using remainder…
- 4x^3 - 12x^2 + 11x - 5 is divided by 2x - 1 Find the remainder using remainder…
- 8x^4 + 12x^3 - 2x^2 - 18x + 14 is divided by x + 1 Find the remainder using…
- x^3 - ax^2 - 5x + 2a is divided by x - a Find the remainder using remainder…
- When the polynomial 2x^3 - 2x^2 + 9x - 8 is divided by x - 3 the remainder is…
- Find the value of m if x^3 - 6x^2 + mx + 60 leaves the remainder 2 when divided…
- If (x - 1) divides mx^3 - 2x^2 + 25x - 26 without remainder find the value of m…
- If the polynomials x^3 + 3x^2 - m and 2x^3 - mx + 9 leaves the same remainder…

**Exercise 3.5**- 6x^4 + 7x^3 - 5x - 4 Determine whether (x + 1) is a factor of the following…
- 2x^4 + 9x^3 + 2x^2 + 10x + 15 Determine whether (x + 1) is a factor of the…
- 3x^3 + 8x^2 + 6x - 5 Determine whether (x + 1) is a factor of the following…
- x^3 - 14x^2 + 3x + 12 Determine whether (x + 1) is a factor of the following…
- Determine whether (x + 4) is a factor of x^3 + 3x^2 - 5x + 36.
- Using factor theorem show that (x - 1) is a factor of 4x^3 - 6x^2 + 9x - 7.…
- Determine whether (2x + 1) is a factor of 4x^3 + 4x^2 - x - 1.
- Determine the value of p if (x + 3) is a factor of x^3 - 3x^2 - px + 24.…

**Exercise 3.6**- The coefficient of x^2 x in 2x^3 - 3x^2 - 2x + 3 are respectively:A. 2, 3 B. -…
- The degree of polynomial 4x^2 - 7x^3 + 6x + 1 is:A. 2 B. 1 C. 3 D. 0…
- The polynomial 3x - 2 is a :A. Linear polynomial B. Quadratic polynomial C.…
- The polynomial 4x^2 + 2x - 2 is a :A. Linear polynomial B. Quadratic polynomial…
- The zero of the polynomial 2x - 5:A. 5/2 B. - 5/2 C. 2/5 D. - 2/5…
- The root of polynomial equation 3x - 1 is:A. - 1/3 B. 1/3 C. 1 D. 3…
- The root of polynomial equation x^2 + 2x = 0:A. 0, 2 B. 1, 2 C. 1, - 2 D. 0, - 2…
- If a polynomial p(x) is divided by (ax + b), then the remainder is:A. p(b/a) B.…
- If a polynomial x^3 - ax^2 + ax - a is divided by (x - a), then the remainder…
- If (ax - b) is a factor of p(x) then,A. p(b) = 0 B. p(- b/a) = 0 C. p(a) = 0 D.…
- One of the factor of x^2 - 3x - 10 is :A. x - 2 B. x + 5 C. x - 5 D. x - 3…
- One of the factor of x^3 - 2x^2 + 2x - 1 is :A. x - 1 B. x + 1 C. x - 2 D. x +…

**Exercise 3.1**

- State whether the following expressions are polynomials in one variable or not.…
- Write the coefficient of x^2 and x in each of the following i. 2 + 3x - 4x^2 +…
- Write the degree of each of the following polynomials. i. 4 - 3x^2 ii. 5y + √2…
- Classify the following polynomials based on their degree. i. 3x^2 + 2x + 1 ii.…
- Give one example of a binomial of degree 27 and monomial of degree 49 and…

**Exercise 3.2**

- Find the zeros of the following polynomials. i. p(x)= 4x - 1 ii. p(x) = 3x + 5…
- Find the roots of the following polynomial equations. i. x - 3 = 0 ii. 5x - 6 =…
- x^2 - 5x + 6 = 0; x = 2, 3 Verify Whether the following are roots of the…
- x^2 + 4x + 3 = 0; x = −1, 2 Verify Whether the following are roots of the…
- x^3 - 2x^2 - 5x + 6 = 0; x = 1, −2, 3 Verify Whether the following are roots of…
- x^3 - 2x^2 - x + 2 = 0; x = −1, 2, 3 Verify Whether the following are roots of…

**Exercise 3.3**

- (5x^3 - 8x^2 + 5x - 7) ÷ (x - 1) Find the quotient the and remainder of the…
- (2x^2 - 3x - 14) ÷ (x + 2) Find the quotient the and remainder of the following…
- (9 + 4x + 5x^2 + 3x^3) ÷ (x+ 1) Find the quotient the and remainder of the…
- (4x^3 - 2x^2 + 6x + 7) ÷ (3 + 2x) Find the quotient the and remainder of the…
- (−18 - 9x + 7x^2) ÷ (x - 2) Find the quotient the and remainder of the…

**Exercise 3.4**

- 3x^3 + 4x^2 - 5x + 8 is divided by x - 1 Find the remainder using remainder…
- 5x^3 + 2x^2 - 6x + 12 is divided by x + 2 Find the remainder using remainder…
- 2x^3 - 4x^2 + 7x + 6 is divided by x - 2 Find the remainder using remainder…
- 4x^3 - 3x^2 + 2x - 4 is divided by x + 3 Find the remainder using remainder…
- 4x^3 - 12x^2 + 11x - 5 is divided by 2x - 1 Find the remainder using remainder…
- 8x^4 + 12x^3 - 2x^2 - 18x + 14 is divided by x + 1 Find the remainder using…
- x^3 - ax^2 - 5x + 2a is divided by x - a Find the remainder using remainder…
- When the polynomial 2x^3 - 2x^2 + 9x - 8 is divided by x - 3 the remainder is…
- Find the value of m if x^3 - 6x^2 + mx + 60 leaves the remainder 2 when divided…
- If (x - 1) divides mx^3 - 2x^2 + 25x - 26 without remainder find the value of m…
- If the polynomials x^3 + 3x^2 - m and 2x^3 - mx + 9 leaves the same remainder…

**Exercise 3.5**

- 6x^4 + 7x^3 - 5x - 4 Determine whether (x + 1) is a factor of the following…
- 2x^4 + 9x^3 + 2x^2 + 10x + 15 Determine whether (x + 1) is a factor of the…
- 3x^3 + 8x^2 + 6x - 5 Determine whether (x + 1) is a factor of the following…
- x^3 - 14x^2 + 3x + 12 Determine whether (x + 1) is a factor of the following…
- Determine whether (x + 4) is a factor of x^3 + 3x^2 - 5x + 36.
- Using factor theorem show that (x - 1) is a factor of 4x^3 - 6x^2 + 9x - 7.…
- Determine whether (2x + 1) is a factor of 4x^3 + 4x^2 - x - 1.
- Determine the value of p if (x + 3) is a factor of x^3 - 3x^2 - px + 24.…

**Exercise 3.6**

- The coefficient of x^2 x in 2x^3 - 3x^2 - 2x + 3 are respectively:A. 2, 3 B. -…
- The degree of polynomial 4x^2 - 7x^3 + 6x + 1 is:A. 2 B. 1 C. 3 D. 0…
- The polynomial 3x - 2 is a :A. Linear polynomial B. Quadratic polynomial C.…
- The polynomial 4x^2 + 2x - 2 is a :A. Linear polynomial B. Quadratic polynomial…
- The zero of the polynomial 2x - 5:A. 5/2 B. - 5/2 C. 2/5 D. - 2/5…
- The root of polynomial equation 3x - 1 is:A. - 1/3 B. 1/3 C. 1 D. 3…
- The root of polynomial equation x^2 + 2x = 0:A. 0, 2 B. 1, 2 C. 1, - 2 D. 0, - 2…
- If a polynomial p(x) is divided by (ax + b), then the remainder is:A. p(b/a) B.…
- If a polynomial x^3 - ax^2 + ax - a is divided by (x - a), then the remainder…
- If (ax - b) is a factor of p(x) then,A. p(b) = 0 B. p(- b/a) = 0 C. p(a) = 0 D.…
- One of the factor of x^2 - 3x - 10 is :A. x - 2 B. x + 5 C. x - 5 D. x - 3…
- One of the factor of x^3 - 2x^2 + 2x - 1 is :A. x - 1 B. x + 1 C. x - 2 D. x +…

###### Exercise 3.1

**Question 1.**State whether the following expressions are polynomials in one variable or not. Give reasons for your answer.

i. 2x^{5} – x^{3} + x – 6

ii. 3x^{2} – 2x + 1

iii. y^{3} + 2√3

iv.

v.

vi. x^{3} + y^{3} + z^{6}

**Answer:**i. 2x^{5} – x^{3} + x – 6

There is only one variable ‘x’ with whole number power. So, this is polynomial in one variable.

ii. 3x^{2} – 2x + 1

There is only one variable ‘x’ with whole number power. So, this is polynomial in one variable.

iii. y^{3} + 2√3

There is only one variable ‘y’ with whole number power. So, this is polynomial in one variable.

iv.

⇒

There is only one variable ‘x’ with whole number power. So, this is polynomial in one variable.

v. 3√ t + 2t

There is only one variable ‘t’ but in 3√t, power of t is which is not a whole number. So, this is not a polynomial in one variable.

vi. x^{3} + y^{3} + z^{3}

There are three variables x, y and z, but power is whole number. So, this is not a polynomial in one variable.

**Question 2.**Write the coefficient of x^{2} and x in each of the following

i. 2 + 3x – 4x^{2} + x^{3}

ii. √3 x + 1

iii. x^{3} + √2 x^{2} + 4x – 1

iv.

**Answer:**i 2 + 3x – 4x^{2} + x^{3}

Co-efficient of x^{2} = -4

Co-efficient of x = 3

ii √3x + 1

Co-efficient of x^{2} = 0

Co-efficient of x = √3

iii x^{3} + √2x^{2} + 4x – 1

Co-efficient of x^{2} = √2

Co-efficient of x = 4

iv

= x^{2} + 3x + 6 = 0

Co-efficient of x^{2} = 1

Co-efficient of x = 3

**Question 3.**Write the degree of each of the following polynomials.

i. 4 – 3x^{2}

ii. 5y + √2

iii. 12 – x + 4x^{3}

iv. 5

**Answer:**i. 4 – 3x^{2}

Degree of the polynomial = 2

ii. 5y + √2

Degree of the polynomial = 1

iii. 12 – x + 4x^{3}

Degree of the polynomial = 3

iv. 5

Degree of the polynomial = 0

**Question 4.**Classify the following polynomials based on their degree.

i. 3x^{2} + 2x + 1

ii. 4x^{3} – 1

iii. y + 3

iv. y^{2} – 4

v. 4x^{3}

vi. 2x

**Answer:**i. 3x^{2} + 2x +1

Since, the highest degree of polynomial is 2

It is a quadratic polynomial.

ii. 4x^{3} – 1

Since, the highest degree of polynomial is 3.

It is a cubic polynomial.

iii. y + 3

Since, the highest degree of polynomial is 1

It is a linear polynomial

iv. y^{2} – 4

Since, the highest degree of polynomial is 2

It is a quadratic polynomial.

v. 4x^{3}

Since, the highest degree of polynomial is 3.

It is a cubic polynomial.

vi. 2x

Since, the highest degree of polynomial is 1

It is a linear polynomial

**Question 5.**Give one example of a binomial of degree 27 and monomial of degree 49 and trinomial of degree 36.

**Answer:**Binomial means having two terms. So binomial of degree 27 is x^{27} + y.

Monomial means having one term. So, monomial of degree is x^{49}.

Trinomial means having three term. So, trinomial of degree is x^{36} + y + 2.

**Question 1.**

State whether the following expressions are polynomials in one variable or not. Give reasons for your answer.

i. 2x^{5} – x^{3} + x – 6

ii. 3x^{2} – 2x + 1

iii. y^{3} + 2√3

iv.

v.

vi. x^{3} + y^{3} + z^{6}

**Answer:**

i. 2x^{5} – x^{3} + x – 6

There is only one variable ‘x’ with whole number power. So, this is polynomial in one variable.

ii. 3x^{2} – 2x + 1

There is only one variable ‘x’ with whole number power. So, this is polynomial in one variable.

iii. y^{3} + 2√3

There is only one variable ‘y’ with whole number power. So, this is polynomial in one variable.

iv.

⇒

There is only one variable ‘x’ with whole number power. So, this is polynomial in one variable.

v. 3√ t + 2t

There is only one variable ‘t’ but in 3√t, power of t is which is not a whole number. So, this is not a polynomial in one variable.

vi. x^{3} + y^{3} + z^{3}

There are three variables x, y and z, but power is whole number. So, this is not a polynomial in one variable.

**Question 2.**

Write the coefficient of x^{2} and x in each of the following

i. 2 + 3x – 4x^{2} + x^{3}

ii. √3 x + 1

iii. x^{3} + √2 x^{2} + 4x – 1

iv.

**Answer:**

i 2 + 3x – 4x^{2} + x^{3}

Co-efficient of x^{2} = -4

Co-efficient of x = 3

ii √3x + 1

Co-efficient of x^{2} = 0

Co-efficient of x = √3

iii x^{3} + √2x^{2} + 4x – 1

Co-efficient of x^{2} = √2

Co-efficient of x = 4

iv

= x^{2} + 3x + 6 = 0

Co-efficient of x^{2} = 1

Co-efficient of x = 3

**Question 3.**

Write the degree of each of the following polynomials.

i. 4 – 3x^{2}

ii. 5y + √2

iii. 12 – x + 4x^{3}

iv. 5

**Answer:**

i. 4 – 3x^{2}

Degree of the polynomial = 2

ii. 5y + √2

Degree of the polynomial = 1

iii. 12 – x + 4x^{3}

Degree of the polynomial = 3

iv. 5

Degree of the polynomial = 0

**Question 4.**

Classify the following polynomials based on their degree.

i. 3x^{2} + 2x + 1

ii. 4x^{3} – 1

iii. y + 3

iv. y^{2} – 4

v. 4x^{3}

vi. 2x

**Answer:**

i. 3x^{2} + 2x +1

Since, the highest degree of polynomial is 2

It is a quadratic polynomial.

ii. 4x^{3} – 1

Since, the highest degree of polynomial is 3.

It is a cubic polynomial.

iii. y + 3

Since, the highest degree of polynomial is 1

It is a linear polynomial

iv. y^{2} – 4

Since, the highest degree of polynomial is 2

It is a quadratic polynomial.

v. 4x^{3}

Since, the highest degree of polynomial is 3.

It is a cubic polynomial.

vi. 2x

Since, the highest degree of polynomial is 1

It is a linear polynomial

**Question 5.**

Give one example of a binomial of degree 27 and monomial of degree 49 and trinomial of degree 36.

**Answer:**

Binomial means having two terms. So binomial of degree 27 is x^{27} + y.

Monomial means having one term. So, monomial of degree is x^{49}.

Trinomial means having three term. So, trinomial of degree is x^{36} + y + 2.

###### Exercise 3.2

**Question 1.**Find the zeros of the following polynomials.

i. p(x)= 4x – 1

ii. p(x) = 3x + 5

iii. p(x) = 2x

v. p(x) = x + 9

**Answer:**i. p(x) = 4x – 1

⇒

⇒

Hence, is the zero of p(x).

ii. p(x) = 3x + 5

⇒

⇒

Hence, is the zero of p(x).

iii. p(x) = 2x

p(0) = 2(0) = 0

Hence, 0 is the zero of p(x).

iv. p(x) = x + 9

p(-9) = -9 + 9 = 0

Hence, -9 is the zero of p(x).

**Question 2.**Find the roots of the following polynomial equations.

i. x – 3 = 0

ii. 5x – 6 = 0

iii. 11x + 1 = 0

iv. −9x = 0

**Answer:**i. x – 5 = 0

⇒ x = 5

∴ x = 5 is a root of x – 5 = 0

ii. 5x – 6 = 0

⇒ 5x = 6

⇒

∴ is a root of 5x – 6 = 0

iii. 11x + 1 = 0

⇒ 11x = -1

⇒

∴ is a root of 11x + 1 = 0.

iv. -9x = 0

⇒

⇒ x = 0

∴ x = 0 is a root of -9x = 0.

**Question 3.**Verify Whether the following are roots of the polynomial equations indicated against them.

x^{2} – 5x + 6 = 0; x = 2, 3

**Answer:**x^{2} – 5x + 6 = 0

p(2) = (2)^{2} – 5(2) + 6

= 4 – 10 + 6

= 10 – 10 = 0

∴ x = 2 is a root of x^{2} – 5x + 6 = 0

p(x) = x^{2} – 5x + 6

p(3) = (3)^{2} – 5(3) + 6

= 9 – 15 + 6

= 15 – 15 = 0

∴ x = 3 is a root of x^{2} – 5x + 6 = 0

**Question 4.**Verify Whether the following are roots of the polynomial equations indicated against them.

x^{2} + 4x + 3 = 0; x = −1, 2

**Answer:**x^{2} + 4x + 3 = 0

let p(x) = x^{2} + 4x + 3

p(-1) = (-1)^{2} + 4(-1) + 3

= 1 – 4 + 3

= 4 – 4 = 0

∴ x = -1 is a root of x^{2} + 4x + 3 = 0

p(x) = x^{2} + 4x + 3 = 0

p(2) = (2)^{2} + 4(2) + 3

= 4 + 8 + 3

= 11 + 4 = 15 ≠ 0

∴ x = 2 is not a root of x^{2} + 4x + 3 = 0.

**Question 5.**Verify Whether the following are roots of the polynomial equations indicated against them.

x^{3} – 2x^{2} – 5x + 6 = 0; x = 1, −2, 3

**Answer:**x^{3} – 2x^{2} – 5x + 6 = 0

let p(x) = x^{3} – 2x^{2} – 5x + 6

p(1) = (1)^{3} – 2(1)^{2} – 5(1) + 6

= 1 – 2 × 1 – 5 + 6

= 1 – 2 – 5 + 6

= 7 – 7 = 0

∴ x = 1 is a root of x^{3} – 2x^{2} – 5x + 6 = 0.

p(x) = x^{3} – 2x^{2} – 5x + 6

p(-2) = (-2)^{3} – 2(-2)^{2} – 5(-2) + 6

= -8 – 2 × 4 – 5 × 2 + 6

= -8 – 8 + 10 + 6

= -16 + 16 = 0

∴ x = -2 is a root of x^{3} – 2x^{2} – 5x + 6 = 0.

p(x) = x^{3} – 2x^{2} – 5x + 6 = 0

p(3) = (3)^{3} – 2(3)^{2} – 5(3) + 6

= 27 – 2 × 9 – 5 × 3 + 6

= 27 – 18 – 15 + 6

= 33 – 33 = 0

∴ x = 3 is a root of x^{3} – 2x^{2} – 5x + 6 = 0.

**Question 6.**Verify Whether the following are roots of the polynomial equations indicated against them.

x^{3} – 2x^{2} – x + 2 = 0; x = −1, 2, 3

**Answer:**x^{3} – 2x^{2} – x + 2 = 0

p(x) = x^{3} – 2x^{2} – x + 2 = 0

p(-1) = (-1)^{3} – 2(-1)^{2} - (-1) + 2

= -1 – 2 × 1 + 1 + 2

= -1 – 2 + 1 + 2

= -3 + 3 = 0

∴ x = -1 is a root of x^{3} – 2x^{2} – x + 2 = 0

p(x) = x^{3} – 2x^{2} – x + 2 = 0

p(2) = (2)^{3} – 2(2)^{2} – (2) + 2

= 8 – 2 × 4 – 2 + 2

= 8 – 8 – 2 + 2

= 10 – 10 = 0

∴ x = 2 is a root of x^{3} – 2x^{2} – x + 2 = 0.

p(x) = x^{3} – 2x^{2} – x + 2 = 0

p(3) = (3)^{3} – 2(3)^{2} – (3) + 2

= 27 – 2 × 9 – 3 + 2

= 27 – 18 – 3 + 2

= 29 – 21 = 8 ≠ 0

∴ x = 3 is not a root of x^{3} – 2x^{2} – x + 2 = 0.

**Question 1.**

Find the zeros of the following polynomials.

i. p(x)= 4x – 1

ii. p(x) = 3x + 5

iii. p(x) = 2x

v. p(x) = x + 9

**Answer:**

i. p(x) = 4x – 1

⇒

⇒

Hence, is the zero of p(x).

ii. p(x) = 3x + 5

⇒

⇒

Hence, is the zero of p(x).

iii. p(x) = 2x

p(0) = 2(0) = 0

Hence, 0 is the zero of p(x).

iv. p(x) = x + 9

p(-9) = -9 + 9 = 0

Hence, -9 is the zero of p(x).

**Question 2.**

Find the roots of the following polynomial equations.

i. x – 3 = 0

ii. 5x – 6 = 0

iii. 11x + 1 = 0

iv. −9x = 0

**Answer:**

i. x – 5 = 0

⇒ x = 5

∴ x = 5 is a root of x – 5 = 0

ii. 5x – 6 = 0

⇒ 5x = 6

⇒

∴ is a root of 5x – 6 = 0

iii. 11x + 1 = 0

⇒ 11x = -1

⇒

∴ is a root of 11x + 1 = 0.

iv. -9x = 0

⇒

⇒ x = 0

∴ x = 0 is a root of -9x = 0.

**Question 3.**

Verify Whether the following are roots of the polynomial equations indicated against them.

x^{2} – 5x + 6 = 0; x = 2, 3

**Answer:**

x^{2} – 5x + 6 = 0

p(2) = (2)^{2} – 5(2) + 6

= 4 – 10 + 6

= 10 – 10 = 0

∴ x = 2 is a root of x^{2} – 5x + 6 = 0

p(x) = x^{2} – 5x + 6

p(3) = (3)^{2} – 5(3) + 6

= 9 – 15 + 6

= 15 – 15 = 0

∴ x = 3 is a root of x^{2} – 5x + 6 = 0

**Question 4.**

Verify Whether the following are roots of the polynomial equations indicated against them.

x^{2} + 4x + 3 = 0; x = −1, 2

**Answer:**

x^{2} + 4x + 3 = 0

let p(x) = x^{2} + 4x + 3

p(-1) = (-1)^{2} + 4(-1) + 3

= 1 – 4 + 3

= 4 – 4 = 0

∴ x = -1 is a root of x^{2} + 4x + 3 = 0

p(x) = x^{2} + 4x + 3 = 0

p(2) = (2)^{2} + 4(2) + 3

= 4 + 8 + 3

= 11 + 4 = 15 ≠ 0

∴ x = 2 is not a root of x^{2} + 4x + 3 = 0.

**Question 5.**

Verify Whether the following are roots of the polynomial equations indicated against them.

x^{3} – 2x^{2} – 5x + 6 = 0; x = 1, −2, 3

**Answer:**

x^{3} – 2x^{2} – 5x + 6 = 0

let p(x) = x^{3} – 2x^{2} – 5x + 6

p(1) = (1)^{3} – 2(1)^{2} – 5(1) + 6

= 1 – 2 × 1 – 5 + 6

= 1 – 2 – 5 + 6

= 7 – 7 = 0

∴ x = 1 is a root of x^{3} – 2x^{2} – 5x + 6 = 0.

p(x) = x^{3} – 2x^{2} – 5x + 6

p(-2) = (-2)^{3} – 2(-2)^{2} – 5(-2) + 6

= -8 – 2 × 4 – 5 × 2 + 6

= -8 – 8 + 10 + 6

= -16 + 16 = 0

∴ x = -2 is a root of x^{3} – 2x^{2} – 5x + 6 = 0.

p(x) = x^{3} – 2x^{2} – 5x + 6 = 0

p(3) = (3)^{3} – 2(3)^{2} – 5(3) + 6

= 27 – 2 × 9 – 5 × 3 + 6

= 27 – 18 – 15 + 6

= 33 – 33 = 0

∴ x = 3 is a root of x^{3} – 2x^{2} – 5x + 6 = 0.

**Question 6.**

Verify Whether the following are roots of the polynomial equations indicated against them.

x^{3} – 2x^{2} – x + 2 = 0; x = −1, 2, 3

**Answer:**

x^{3} – 2x^{2} – x + 2 = 0

p(x) = x^{3} – 2x^{2} – x + 2 = 0

p(-1) = (-1)^{3} – 2(-1)^{2} - (-1) + 2

= -1 – 2 × 1 + 1 + 2

= -1 – 2 + 1 + 2

= -3 + 3 = 0

∴ x = -1 is a root of x^{3} – 2x^{2} – x + 2 = 0

p(x) = x^{3} – 2x^{2} – x + 2 = 0

p(2) = (2)^{3} – 2(2)^{2} – (2) + 2

= 8 – 2 × 4 – 2 + 2

= 8 – 8 – 2 + 2

= 10 – 10 = 0

∴ x = 2 is a root of x^{3} – 2x^{2} – x + 2 = 0.

p(x) = x^{3} – 2x^{2} – x + 2 = 0

p(3) = (3)^{3} – 2(3)^{2} – (3) + 2

= 27 – 2 × 9 – 3 + 2

= 27 – 18 – 3 + 2

= 29 – 21 = 8 ≠ 0

∴ x = 3 is not a root of x^{3} – 2x^{2} – x + 2 = 0.

###### Exercise 3.3

**Question 1.**Find the quotient the and remainder of the following division.

(5x^{3} – 8x^{2} + 5x – 7) ÷ (x – 1)

**Answer:**(5x^{3} – 8x^{2} + 5x – 7) ÷ (x – 1)

We see that the equation is already arranged in descending order.

Now we need to divide (5x^{3} – 8x^{2} + 5x – 7) by (x – 1).

Now we need to find out by how much should we multiple “x” to get a value as much as 5x^{3}.

To get x^{3}, we need to multiply x×x^{2}.

Therefore, we need to multiply with 5x^{2} × (x – 1) and we get (5x^{3} – 5x^{2}) now subtract (5x^{3} – 5x^{2}) from 5x^{3} – 8x^{2} + 5x – 7 so we get – 3x^{2}.

Now we carry 5x – 7 along with – 3x^{2}, as shown below

So, in same way we have keep dividing till we get rid of x as shown below.

here (x – 1) × (– 3x)

= – 3x^{2} + 3x

here (x – 1) × 2

= 2x – 2

Therefore, we got the quotient = 5x^{2} – 3x + 2 and

Remainder = –5

**Question 2.**Find the quotient the and remainder of the following division.

(2x^{2} – 3x – 14) ÷ (x + 2)

**Answer:**(2x^{2} – 3x – 14) ÷ (x + 2)

We see that the equation is already arranged in descending order.

Now we need to divide (2x^{2} – 3x – 14) by (x + 2).

Now we need to find out by how much should we multiple “x” to get a value as much as 2x^{2}.

To get x^{2}, we need to multiply x×x.

Therefore, we need to multiply with 2x × (x + 2) and we get (2x^{2} + 4x) now subtract (2x^{2} + 4x) from 2x^{2} – 3x – 14 so we get –7x.

Now we carry 14 along with –7x, as shown below

So, in same way we have keep dividing till we get rid of x as shown below.

here (x + 2) × (– 7)

= – 7x – 14

Therefore, we got the quotient = 2x – 7 and

Remainder = 0

**Question 3.**Find the quotient the and remainder of the following division.

(9 + 4x + 5x^{2} + 3x^{3}) ÷ (x+ 1)

**Answer:**(9 + 4x + 5x^{2} + 3x^{3}) ÷ (x + 1)

We see that the equation is not arranged in descending order, so we need first arrange it in descending order of the power of x.

Therefore it becomes,

(3x^{3} + 5x^{2} + 4x + 9) ÷ (x + 1)

Now we need to divide (3x^{3} + 5x^{2} + 4x + 9) by (x + 1).

Now we need to find out by how much should we multiple “x” to get a value as much as 3x^{3}.

To get x^{3}, we need to multiply x×x^{2}.

Therefore, we need to multiply with 3x^{2} × (x + 1) and we get (3x^{3} + 3x^{2}) now subtract (3x^{3} + 3x^{2}) from 3x^{3} + 5x^{2} + 4x + 9 so we get 2x^{2}.

Now we carry 4x + 9 along with 2x^{2}, as shown below

So, in same way we have keep dividing till we get rid of x as shown below.

here (x + 1) × ( 2x)

= 2x^{2} + 2x

here (x + 1) × 2

= 2x + 2

Therefore, we got the quotient = 3x^{2} + 2x + 2 and

Remainder = 7

**Question 4.**Find the quotient the and remainder of the following division.

(4x^{3} – 2x^{2} + 6x + 7) ÷ (3 + 2x)

**Answer:**(4x^{3} – 2x^{2} + 6x + 7) ÷ (3 + 2x)

We see that the equation is not arranged in descending order, so we need first arrange it in descending order of the power of x.

Therefore it becomes,

(4x^{3} – 2x^{2} + 6x + 7) ÷ (2x + 3)

Now we need to divide (4x^{3} – 2x^{2} + 6x + 7) by (2x + 3)

Now we need to find out by how much should we multiple “x” to get a value as much as 4x^{3}.

To get x^{3}, we need to multiply x×x^{2}.

Therefore we need to multiply with 2x^{2} × (2x + 3) and we get (4x^{3} + 6x^{2}) now subtract (4x^{3} + 6x^{2}) from 4x^{3} – 2x^{2} + 6x + 7so we get – 8x^{2}.

Now we carry 6x + 7along with 4x^{2}, as shown below

So, in same way we have keep dividing till we get rid of x as shown below.

here (2x + 3) × (–4x)

= – 8x^{2} – 12x

here (2x + 3) × 9

= 18x + 27

Therefore, we got the quotient = 2x^{2} – 4x + 9 and

Remainder = –20

**Question 5.**Find the quotient the and remainder of the following division.

(−18 – 9x + 7x^{2}) ÷ (x – 2)

**Answer:**(−18 – 9x + 7x^{2}) ÷ (x – 2)

We see that the equation is not arranged in descending order, so we need first arrange it in descending order of the power of x.

Therefore it becomes,

(7x^{2} – 9x – 18) ÷ (x – 2)

Now we need to divide (7x^{2} – 9x – 18) by (x – 2).

Now we need to find out by how much should we multiple “x” to get a value as much as 7x^{2}.

To get x^{2}, we need to multiply x×x.

Therefore, we need to multiply with 7x × (x – 2) and we get (7x^{2} – 14x) now subtract (7x^{2} – 14x) from 7x^{2} – 9x – 18 so we get 5x.

Now we carry 18 along with 5x, as shown below

So, in same way we have keep dividing till we get rid of x as shown below.

here (x – 2) × 5

= 5x – 10

Therefore, we got the quotient = 7x + 5 and

Remainder = – 8

**Question 1.**

Find the quotient the and remainder of the following division.

(5x^{3} – 8x^{2} + 5x – 7) ÷ (x – 1)

**Answer:**

(5x^{3} – 8x^{2} + 5x – 7) ÷ (x – 1)

We see that the equation is already arranged in descending order.

Now we need to divide (5x^{3} – 8x^{2} + 5x – 7) by (x – 1).

Now we need to find out by how much should we multiple “x” to get a value as much as 5x^{3}.

To get x^{3}, we need to multiply x×x^{2}.

Therefore, we need to multiply with 5x^{2} × (x – 1) and we get (5x^{3} – 5x^{2}) now subtract (5x^{3} – 5x^{2}) from 5x^{3} – 8x^{2} + 5x – 7 so we get – 3x^{2}.

Now we carry 5x – 7 along with – 3x^{2}, as shown below

So, in same way we have keep dividing till we get rid of x as shown below.

here (x – 1) × (– 3x)

= – 3x^{2} + 3x

here (x – 1) × 2

= 2x – 2

Therefore, we got the quotient = 5x^{2} – 3x + 2 and

Remainder = –5

**Question 2.**

Find the quotient the and remainder of the following division.

(2x^{2} – 3x – 14) ÷ (x + 2)

**Answer:**

(2x^{2} – 3x – 14) ÷ (x + 2)

We see that the equation is already arranged in descending order.

Now we need to divide (2x^{2} – 3x – 14) by (x + 2).

Now we need to find out by how much should we multiple “x” to get a value as much as 2x^{2}.

To get x^{2}, we need to multiply x×x.

Therefore, we need to multiply with 2x × (x + 2) and we get (2x^{2} + 4x) now subtract (2x^{2} + 4x) from 2x^{2} – 3x – 14 so we get –7x.

Now we carry 14 along with –7x, as shown below

So, in same way we have keep dividing till we get rid of x as shown below.

here (x + 2) × (– 7)

= – 7x – 14

Therefore, we got the quotient = 2x – 7 and

Remainder = 0

**Question 3.**

Find the quotient the and remainder of the following division.

(9 + 4x + 5x^{2} + 3x^{3}) ÷ (x+ 1)

**Answer:**

(9 + 4x + 5x^{2} + 3x^{3}) ÷ (x + 1)

We see that the equation is not arranged in descending order, so we need first arrange it in descending order of the power of x.

Therefore it becomes,

(3x^{3} + 5x^{2} + 4x + 9) ÷ (x + 1)

Now we need to divide (3x^{3} + 5x^{2} + 4x + 9) by (x + 1).

Now we need to find out by how much should we multiple “x” to get a value as much as 3x^{3}.

To get x^{3}, we need to multiply x×x^{2}.

Therefore, we need to multiply with 3x^{2} × (x + 1) and we get (3x^{3} + 3x^{2}) now subtract (3x^{3} + 3x^{2}) from 3x^{3} + 5x^{2} + 4x + 9 so we get 2x^{2}.

Now we carry 4x + 9 along with 2x^{2}, as shown below

So, in same way we have keep dividing till we get rid of x as shown below.

here (x + 1) × ( 2x)

= 2x^{2} + 2x

here (x + 1) × 2

= 2x + 2

Therefore, we got the quotient = 3x^{2} + 2x + 2 and

Remainder = 7

**Question 4.**

Find the quotient the and remainder of the following division.

(4x^{3} – 2x^{2} + 6x + 7) ÷ (3 + 2x)

**Answer:**

(4x^{3} – 2x^{2} + 6x + 7) ÷ (3 + 2x)

We see that the equation is not arranged in descending order, so we need first arrange it in descending order of the power of x.

Therefore it becomes,

(4x^{3} – 2x^{2} + 6x + 7) ÷ (2x + 3)

Now we need to divide (4x^{3} – 2x^{2} + 6x + 7) by (2x + 3)

Now we need to find out by how much should we multiple “x” to get a value as much as 4x^{3}.

To get x^{3}, we need to multiply x×x^{2}.

Therefore we need to multiply with 2x^{2} × (2x + 3) and we get (4x^{3} + 6x^{2}) now subtract (4x^{3} + 6x^{2}) from 4x^{3} – 2x^{2} + 6x + 7so we get – 8x^{2}.

Now we carry 6x + 7along with 4x^{2}, as shown below

So, in same way we have keep dividing till we get rid of x as shown below.

here (2x + 3) × (–4x)

= – 8x^{2} – 12x

here (2x + 3) × 9

= 18x + 27

Therefore, we got the quotient = 2x^{2} – 4x + 9 and

Remainder = –20

**Question 5.**

Find the quotient the and remainder of the following division.

(−18 – 9x + 7x^{2}) ÷ (x – 2)

**Answer:**

(−18 – 9x + 7x^{2}) ÷ (x – 2)

We see that the equation is not arranged in descending order, so we need first arrange it in descending order of the power of x.

Therefore it becomes,

(7x^{2} – 9x – 18) ÷ (x – 2)

Now we need to divide (7x^{2} – 9x – 18) by (x – 2).

Now we need to find out by how much should we multiple “x” to get a value as much as 7x^{2}.

To get x^{2}, we need to multiply x×x.

Therefore, we need to multiply with 7x × (x – 2) and we get (7x^{2} – 14x) now subtract (7x^{2} – 14x) from 7x^{2} – 9x – 18 so we get 5x.

Now we carry 18 along with 5x, as shown below

So, in same way we have keep dividing till we get rid of x as shown below.

here (x – 2) × 5

= 5x – 10

Therefore, we got the quotient = 7x + 5 and

Remainder = – 8

###### Exercise 3.4

**Question 1.**Find the remainder using remainder theorem, when

3x^{3} + 4x^{2} – 5x + 8 is divided by x – 1

**Answer:**3x^{3} + 4x^{2} – 5x + 8 is divided by x – 1

Remainder theorem states that if p(x) is any polynomial and a is any real number and If p(x) is divided by the linear polynomial (x – a) , then the remainder is p(a).

Let p(x) = 3x^{3} + 4x^{2} – 5x + 8 and we have (x – 1)

The zero of (x – 1) is 1

Now using Remainder theorem,

p(x) = 3x^{3} + 4x^{2} – 5x + 8 is divided by x – 1 then, p(1) is the remainder

p(1) = 3(1)^{3} + 4(1)^{2} – 5(1) + 8

= 3 + 4 – 5 + 8

= 10

Remainder = 10

**Question 2.**Find the remainder using remainder theorem, when

5x^{3} + 2x^{2} – 6x + 12 is divided by x + 2

**Answer:**5x^{3} + 2x^{2} – 6x + 12 is divided by x + 2

Remainder theorem states that if p(x) is any polynomial and a is any real number and If p(x) is divided by the linear polynomial (x – a) , then the remainder is p(a).

Let p(x) = 5x^{3} + 2x^{2} – 6x + 12 and we have (x + 2)

The zero of (x + 2) is – 2

Now using Remainder theorem,

p(x) = 5x^{3} + 2x^{2} – 6x + 12 is divided by x + 2 then, p(–2) is the remainder

p(–2) = 5(–2)^{3} + 2(–2)^{2} – 6(–2) + 12

= 5×(–8) + 2×4 – (– 12) + 12

= – 40 + 8 + 12 + 12

= – 40 + 32

= – 8

Remainder = –8

**Question 3.**Find the remainder using remainder theorem, when

2x^{3} – 4x^{2} + 7x + 6 is divided by x – 2

**Answer:**2x^{3} – 4x^{2} + 7x + 6 is divided by x – 2

Remainder theorem states that if p(x) is any polynomial and a is any real number and If p(x) is divided by the linear polynomial (x – a) , then the remainder is p(a).

Let p(x) = 2x^{3} – 4x^{2} + 7x + 6 and we have (x – 2)

The zero of (x – 2) is 2

Now using Remainder theorem,

p(x) = 2x^{3} – 4x^{2} + 7x + 6 is divided by x – 2 then, p(2) is the remainder

p(2) = 2(2)^{3} – 4(2)^{2} + 7(2) + 6

= 16 – 16 + 14 +6

= 20

Remainder = 20

**Question 4.**Find the remainder using remainder theorem, when

4x^{3} – 3x^{2} + 2x – 4 is divided by x + 3

**Answer:**4x^{3} – 3x^{2} + 2x – 4 is divided by x + 3

Remainder theorem states that if p(x) is any polynomial and a is any real number and If p(x) is divided by the linear polynomial (x – a) , then the remainder is p(a).

Let p(x) = 4x^{3} – 3x^{2} + 2x – 4 and we have (x + 3)

The zero of (x + 3) is – 3

Now using Remainder theorem,

p(x) = 4x^{3} – 3x^{2} + 2x – 4 is divided by x + 3 then, p(– 3) is the remainder

p(– 3) = 4(–3)^{3} – 3(–3)^{2} + 2(–3) – 4

= – 108 – 27 – 6 – 4

= – 145

Remainder = –145

**Question 5.**Find the remainder using remainder theorem, when

4x^{3} – 12x^{2} + 11x – 5 is divided by 2x – 1

**Answer:**4x^{3} – 12x^{2} + 11x – 5 is divided by 2x – 1

Remainder theorem states that if p(x) is any polynomial and a is any real number and If p(x) is divided by the linear polynomial (x – a) , then the remainder is p(a).

Let p(x) = 4x^{3} – 12x^{2} + 11x – 5 and we have (2x – 1)

The zero of (2x – 1) is

Now using Remainder theorem,

p(x) = 4x^{3} – 12x^{2} + 11x – 5 is divided by 2x – 1 then, is the remainder

Remainder = –2

**Question 6.**Find the remainder using remainder theorem, when

8x^{4} + 12x^{3} – 2x^{2} – 18x + 14 is divided by x + 1

**Answer:**8x^{4} + 12x^{3} – 2x^{2} – 18x + 14 is divided by x + 1

Remainder theorem states that if p(x) is any polynomial and a is any real number and If p(x) is divided by the linear polynomial (x – a) , then the remainder is p(a).

Let p(x) = 8x^{4} + 12x^{3} – 2x^{2} – 18x + 14 and we have (x + 1)

The zero of (x + 1) is – 1

Now using Remainder theorem,

p(x) = 8x^{4} + 12x^{3} – 2x^{2} – 18x + 14 is divided by x + 1 then, p(– 1) is the remainder

p(– 1) = 8(–1)^{4} + 12(–1)^{3} – 2(–1)^{2} – 18(–1) + 14

= 8 – 12 – 2 +18 + 14

= 26

Remainder = 26

**Question 7.**Find the remainder using remainder theorem, when

x^{3} – ax^{2} – 5x + 2a is divided by x – a

**Answer:**x^{3} – ax^{2} – 5x + 2a is divided by x – a

Remainder theorem states that if p(x) is any polynomial and a is any real number and If p(x) is divided by the linear polynomial (x – a) , then the remainder is p(a).

Let p(x) = x^{3} – ax^{2} – 5x + 2a and we have (x – a)

The zero of (x – a) is a

Now using Remainder theorem,

p(x) = x^{3} – ax^{2} – 5x + 2a is divided by x – a then, p(a) is the remainder

p(a) = (a)^{3} – a(a)^{2} – 5(a) + 2a

= a^{3} – a^{3} – 5a+ 2a

= – 3a

Remainder = –3a

**Question 8.**When the polynomial 2x^{3} – 2x^{2} + 9x – 8 is divided by x – 3 the remainder is 28. Find the value of a.

**Answer:**2x^{3} – ax^{2} + 9x – 8 is divided by x – 3 and remainder = 28

Remainder theorem states that if p(x) is any polynomial and a is any real number and If p(x) is divided by the linear polynomial (x – a) , then the remainder is p(a).

Let p(x) = 2x^{3} – ax^{2} + 9x – 8 and we have (x – 3)

The zero of (x – 3) is 3

Now using Remainder theorem,

p(x) = 2x^{3} – ax^{2} + 9x – 8 is divided by x – a then, p(3) is the remainder which is 28

p(3) = 2x^{3} – ax^{2} + 9x – 8 =28

= 2(3)^{3} – a(3)^{2} + 9(3) – 8 =28

= 54 – 9a + 27 – 8 = 28

= 73 – 9a = 28

= 9a = 73 –28

= 9a = 45

a = 5

**Question 9.**Find the value of m if x^{3} – 6x^{2} + mx + 60 leaves the remainder 2 when divided by (x + 2).

**Answer:**x^{3} – 6x^{2} + mx + 60 divided by (x + 2) and remainder = 2

Remainder theorem states that if p(x) is any polynomial and a is any real number and If p(x) is divided by the linear polynomial (x – a) , then the remainder is p(a).

Let p(x) = x^{3} – 6x^{2} + mx + 60 and we have (x + 2)

The zero of (x + 2) is –2

Now using Remainder theorem,

p(x) = x^{3} – 6x^{2} + mx + 60 is divided by x + 2 then, p(–2) is the remainder which is 2

p(–2) = x^{3} – 6x^{2} + mx + 60 = 2

= (–2)^{3} – 6(–2)^{2} + m(–2) + 60 =2

= – 8 – 24 – 2m + 60 = 2

= – 32 – 2m + 60 = 2

= 28 – 2m = 2

= 2m = 28 – 2

= 2m = 26

m = 13

**Question 10.**If (x – 1) divides mx^{3} – 2x^{2} + 25x – 26 without remainder find the value of m

**Answer:**mx^{3} – 2x^{2} + 25x – 26 is divided by (x – 1) without remainder that means remainder = 0

Remainder theorem states that if p(x) is any polynomial and a is any real number and If p(x) is divided by the linear polynomial (x – a) , then the remainder is p(a).

Let p(x) mx^{3} – 2x^{2} + 25x – 26 and we have (x – 1)

The zero of (x – 1) is 1

Now using Remainder theorem,

p(x) = mx^{3} – 2x^{2} + 25x – 26 is divided by x – 1 then, p(1) is the remainder which is 0

p(1) = mx^{3} – 2x^{2} + 25x – 26 = 0

= m(1)^{3} – 2(1)^{2} + 25(1) – 26 = 0

= m – 2 + 25 – 26 = 0

= m – 3 = 0

m = 3

**Question 11.**If the polynomials x^{3} + 3x^{2} – m and 2x^{3} – mx + 9 leaves the same remainder when they are divided by (x – 2), find the value of m. Also find the remainder

**Answer:**x^{3} + 3x^{2} – m and 2x^{3} – mx + 9 is divided by (x – 2) and the remainder is same.

Now let p(x) = x^{3} + 3x^{2} – m is divided by x – 2 then, p(2) is the remainder

p(2) = (2)^{3} + 3(2)^{2} – m

= 8 + 12 – m

= 20 – m

Now let q(x) = 2x^{3} – mx + 9 is divided by x – 2 then, q(2) is the remainder

q(2) = 2(2)^{3} – m(2) + 9

= 16 – 2m + 9

= 25 – 2m

Now, as the question says that the remainder for p(x) and q(x) is same

Therefore, p(2) = q(2)

20 – m = 25 – 2m

2m – m = 25 – 20

m = 5

Remainder = p(2) = 20 – m

= 15

**Question 1.**

Find the remainder using remainder theorem, when

3x^{3} + 4x^{2} – 5x + 8 is divided by x – 1

**Answer:**

3x^{3} + 4x^{2} – 5x + 8 is divided by x – 1

Remainder theorem states that if p(x) is any polynomial and a is any real number and If p(x) is divided by the linear polynomial (x – a) , then the remainder is p(a).

Let p(x) = 3x^{3} + 4x^{2} – 5x + 8 and we have (x – 1)

The zero of (x – 1) is 1

Now using Remainder theorem,

p(x) = 3x^{3} + 4x^{2} – 5x + 8 is divided by x – 1 then, p(1) is the remainder

p(1) = 3(1)^{3} + 4(1)^{2} – 5(1) + 8

= 3 + 4 – 5 + 8

= 10

Remainder = 10

**Question 2.**

Find the remainder using remainder theorem, when

5x^{3} + 2x^{2} – 6x + 12 is divided by x + 2

**Answer:**

5x^{3} + 2x^{2} – 6x + 12 is divided by x + 2

Remainder theorem states that if p(x) is any polynomial and a is any real number and If p(x) is divided by the linear polynomial (x – a) , then the remainder is p(a).

Let p(x) = 5x^{3} + 2x^{2} – 6x + 12 and we have (x + 2)

The zero of (x + 2) is – 2

Now using Remainder theorem,

p(x) = 5x^{3} + 2x^{2} – 6x + 12 is divided by x + 2 then, p(–2) is the remainder

p(–2) = 5(–2)^{3} + 2(–2)^{2} – 6(–2) + 12

= 5×(–8) + 2×4 – (– 12) + 12

= – 40 + 8 + 12 + 12

= – 40 + 32

= – 8

Remainder = –8

**Question 3.**

Find the remainder using remainder theorem, when

2x^{3} – 4x^{2} + 7x + 6 is divided by x – 2

**Answer:**

2x^{3} – 4x^{2} + 7x + 6 is divided by x – 2

Remainder theorem states that if p(x) is any polynomial and a is any real number and If p(x) is divided by the linear polynomial (x – a) , then the remainder is p(a).

Let p(x) = 2x^{3} – 4x^{2} + 7x + 6 and we have (x – 2)

The zero of (x – 2) is 2

Now using Remainder theorem,

p(x) = 2x^{3} – 4x^{2} + 7x + 6 is divided by x – 2 then, p(2) is the remainder

p(2) = 2(2)^{3} – 4(2)^{2} + 7(2) + 6

= 16 – 16 + 14 +6

= 20

Remainder = 20

**Question 4.**

Find the remainder using remainder theorem, when

4x^{3} – 3x^{2} + 2x – 4 is divided by x + 3

**Answer:**

4x^{3} – 3x^{2} + 2x – 4 is divided by x + 3

Let p(x) = 4x^{3} – 3x^{2} + 2x – 4 and we have (x + 3)

The zero of (x + 3) is – 3

Now using Remainder theorem,

p(x) = 4x^{3} – 3x^{2} + 2x – 4 is divided by x + 3 then, p(– 3) is the remainder

p(– 3) = 4(–3)^{3} – 3(–3)^{2} + 2(–3) – 4

= – 108 – 27 – 6 – 4

= – 145

Remainder = –145

**Question 5.**

Find the remainder using remainder theorem, when

4x^{3} – 12x^{2} + 11x – 5 is divided by 2x – 1

**Answer:**

4x^{3} – 12x^{2} + 11x – 5 is divided by 2x – 1

Let p(x) = 4x^{3} – 12x^{2} + 11x – 5 and we have (2x – 1)

The zero of (2x – 1) is

Now using Remainder theorem,

p(x) = 4x^{3} – 12x^{2} + 11x – 5 is divided by 2x – 1 then, is the remainder

Remainder = –2

**Question 6.**

Find the remainder using remainder theorem, when

8x^{4} + 12x^{3} – 2x^{2} – 18x + 14 is divided by x + 1

**Answer:**

8x^{4} + 12x^{3} – 2x^{2} – 18x + 14 is divided by x + 1

Let p(x) = 8x^{4} + 12x^{3} – 2x^{2} – 18x + 14 and we have (x + 1)

The zero of (x + 1) is – 1

Now using Remainder theorem,

p(x) = 8x^{4} + 12x^{3} – 2x^{2} – 18x + 14 is divided by x + 1 then, p(– 1) is the remainder

p(– 1) = 8(–1)^{4} + 12(–1)^{3} – 2(–1)^{2} – 18(–1) + 14

= 8 – 12 – 2 +18 + 14

= 26

Remainder = 26

**Question 7.**

Find the remainder using remainder theorem, when

x^{3} – ax^{2} – 5x + 2a is divided by x – a

**Answer:**

x^{3} – ax^{2} – 5x + 2a is divided by x – a

Let p(x) = x^{3} – ax^{2} – 5x + 2a and we have (x – a)

The zero of (x – a) is a

Now using Remainder theorem,

p(x) = x^{3} – ax^{2} – 5x + 2a is divided by x – a then, p(a) is the remainder

p(a) = (a)^{3} – a(a)^{2} – 5(a) + 2a

= a^{3} – a^{3} – 5a+ 2a

= – 3a

Remainder = –3a

**Question 8.**

When the polynomial 2x^{3} – 2x^{2} + 9x – 8 is divided by x – 3 the remainder is 28. Find the value of a.

**Answer:**

2x^{3} – ax^{2} + 9x – 8 is divided by x – 3 and remainder = 28

Let p(x) = 2x^{3} – ax^{2} + 9x – 8 and we have (x – 3)

The zero of (x – 3) is 3

Now using Remainder theorem,

p(x) = 2x^{3} – ax^{2} + 9x – 8 is divided by x – a then, p(3) is the remainder which is 28

p(3) = 2x^{3} – ax^{2} + 9x – 8 =28

= 2(3)^{3} – a(3)^{2} + 9(3) – 8 =28

= 54 – 9a + 27 – 8 = 28

= 73 – 9a = 28

= 9a = 73 –28

= 9a = 45

a = 5

**Question 9.**

Find the value of m if x^{3} – 6x^{2} + mx + 60 leaves the remainder 2 when divided by (x + 2).

**Answer:**

x^{3} – 6x^{2} + mx + 60 divided by (x + 2) and remainder = 2

Let p(x) = x^{3} – 6x^{2} + mx + 60 and we have (x + 2)

The zero of (x + 2) is –2

Now using Remainder theorem,

p(x) = x^{3} – 6x^{2} + mx + 60 is divided by x + 2 then, p(–2) is the remainder which is 2

p(–2) = x^{3} – 6x^{2} + mx + 60 = 2

= (–2)^{3} – 6(–2)^{2} + m(–2) + 60 =2

= – 8 – 24 – 2m + 60 = 2

= – 32 – 2m + 60 = 2

= 28 – 2m = 2

= 2m = 28 – 2

= 2m = 26

m = 13

**Question 10.**

If (x – 1) divides mx^{3} – 2x^{2} + 25x – 26 without remainder find the value of m

**Answer:**

mx^{3} – 2x^{2} + 25x – 26 is divided by (x – 1) without remainder that means remainder = 0

Let p(x) mx^{3} – 2x^{2} + 25x – 26 and we have (x – 1)

The zero of (x – 1) is 1

Now using Remainder theorem,

p(x) = mx^{3} – 2x^{2} + 25x – 26 is divided by x – 1 then, p(1) is the remainder which is 0

p(1) = mx^{3} – 2x^{2} + 25x – 26 = 0

= m(1)^{3} – 2(1)^{2} + 25(1) – 26 = 0

= m – 2 + 25 – 26 = 0

= m – 3 = 0

m = 3

**Question 11.**

If the polynomials x^{3} + 3x^{2} – m and 2x^{3} – mx + 9 leaves the same remainder when they are divided by (x – 2), find the value of m. Also find the remainder

**Answer:**

x^{3} + 3x^{2} – m and 2x^{3} – mx + 9 is divided by (x – 2) and the remainder is same.

Now let p(x) = x^{3} + 3x^{2} – m is divided by x – 2 then, p(2) is the remainder

p(2) = (2)^{3} + 3(2)^{2} – m

= 8 + 12 – m

= 20 – m

Now let q(x) = 2x^{3} – mx + 9 is divided by x – 2 then, q(2) is the remainder

q(2) = 2(2)^{3} – m(2) + 9

= 16 – 2m + 9

= 25 – 2m

Now, as the question says that the remainder for p(x) and q(x) is same

Therefore, p(2) = q(2)

20 – m = 25 – 2m

2m – m = 25 – 20

m = 5

Remainder = p(2) = 20 – m

= 15

###### Exercise 3.5

**Question 1.**Determine whether (x + 1) is a factor of the following polynomials:

6x^{4} + 7x^{3} – 5x – 4

**Answer:**Let f(x) = 6x^{4} + 7x^{3} – 5x – 4

By factor theorem,

x + 1 = 0 ;x = – 1

If f(– 1) = 0 then (x + 1) is a factor of f(x)

∴f(– 1) = 6(– 1)^{4} + 7(– 1)^{3} – 5(– 1) – 4

= 6 – 7 + 5 – 4 = 11 – 11 = 0

∴(x + 1) is a factor of f(x) = 6x^{4} + 7x^{3} – 5x – 4

**Question 2.**Determine whether (x + 1) is a factor of the following polynomials:

2x^{4} + 9x^{3} + 2x^{2} + 10x + 15

**Answer:**Let f(x) = 2x^{4} + 9x^{3} + 2x^{2} + 10x + 15

By factor theorem,

x + 1 = 0 ;x = – 1

If f(– 1) = 0 then (x + 1) is a factor of f(x)

∴f(– 1) = 2(– 1)^{4} + 9(– 1)^{3} + 2(– 1)^{2} + 10(– 1) + 15

= 2 – 9 + 2 – 10 + 15 = 19 – 19 = 0

∴(x + 1) is a factor of f(x) = 2x^{4} + 9x^{3} + 2x^{2} + 10x + 15

**Question 3.**Determine whether (x + 1) is a factor of the following polynomials:

3x^{3} + 8x^{2} + 6x – 5

**Answer:**Let f(x) = 3x^{3} + 8x^{2} + 6x – 5

By factor theorem,

x + 1 = 0 ;x = – 1

If f(– 1) = 0 then (x + 1) is a factor of f(x)

∴f(– 1) = 3(– 1)^{3} + 8(– 1)^{2} – 6(– 1) – 5

= – 3 + 8 + 6 – 5 = 6(not equal to 0)

∴(x + 1) is not a factor of f(x) = 3x^{3} + 8x^{2} + 6x – 5

**Question 4.**Determine whether (x + 1) is a factor of the following polynomials:

x^{3} – 14x^{2} + 3x + 12

**Answer:**Let f(x) = x^{3} – 14x^{2} + 3x + 12

By factor theorem,

x + 1 = 0 ;x = – 1

If f(– 1) = 0 then (x + 1) is a factor of f(x)

∴f(– 1) = (– 1)^{3} – 14(– 1)^{2} + 3(– 1) + 12

= – 1 – 14 – 3 + 12 = – 6(not equal to 0)

∴(x + 1) is not a factor of f(x) = x^{3} – 14x^{2} + 3x + 12

**Question 5.**Determine whether (x + 4) is a factor of x^{3} + 3x^{2} – 5x + 36.

**Answer:**Let f(x) = x^{3} + 3x^{2} – 5x + 36.

By factor theorem,

x + 4 = 0: x = – 4

If f(– 4) = 0, then (x + 4) is a factor

∴f(– 4) = (– 4)^{3} + 3(– 4)^{2} – 5(– 4) + 36

= – 64 + 48 + 20 + 36

= – 64 + 104 = 40

∴f(– 4) is not equal to 0

So, (x + 4) is not a factor of f(x).

**Question 6.**Using factor theorem show that (x – 1) is a factor of 4x^{3} – 6x^{2} + 9x – 7.

**Answer:**f(x) = 4x^{3} – 6x^{2} + 9x – 7

By factor theorem,

(x – 1) = 0 ; x = 1

Since, (x – 1) is a factor of f(x)

Therefore, f(1) = 0

f(1) = 4(1)^{3} – 6(1)^{2} + 9(1) – 7 = 4 – 6 + 9 – 7 = 13 – 13 = 0

∴(x – 1) is a factor of f(x)

**Question 7.**Determine whether (2x + 1) is a factor of 4x^{3} + 4x^{2} – x – 1.

**Answer:**Let f(x) = 4x^{3} + 4x^{2} – x – 1

By factor Theorem,

2x + 1 = 0 ; x = – 1/2

∴f(– 1/2) = 4(– 1/2)^{3} + 4(– 1/2)^{2} – (– 1/2) – 1

= 4(– 1/8) + 4(1/4) + (1/2) – 1

= (– 1/2) + 1 + (1/2) – 1 = 0

∴f(– 1/2) = 0

So, (2x + 1) is a factor of f(x).

**Question 8.**Determine the value of p if (x + 3) is a factor of x^{3} – 3x^{2} – px + 24.

**Answer:**Let f(x) = x^{3} + 3x^{2} – px + 24.

By factor theorem,

x + 3 = 0;x = – 3

∴(x + 3) is a factor of f(x)

So, f(– 3) = 0.

f(– 3) = (– 3)^{3} – 3(– 3)^{2} – p(– 3) + 24 = 0

= >27 – 27 + 3p + 24 = 0

= > – 59 + 24 + 3p = 0

∴3p – 30 = 0

⇒ p = 30/3

**⇒****p = 10**

**Question 1.**

Determine whether (x + 1) is a factor of the following polynomials:

6x^{4} + 7x^{3} – 5x – 4

**Answer:**

Let f(x) = 6x^{4} + 7x^{3} – 5x – 4

By factor theorem,

x + 1 = 0 ;x = – 1

If f(– 1) = 0 then (x + 1) is a factor of f(x)

∴f(– 1) = 6(– 1)^{4} + 7(– 1)^{3} – 5(– 1) – 4

= 6 – 7 + 5 – 4 = 11 – 11 = 0

∴(x + 1) is a factor of f(x) = 6x^{4} + 7x^{3} – 5x – 4

**Question 2.**

Determine whether (x + 1) is a factor of the following polynomials:

2x^{4} + 9x^{3} + 2x^{2} + 10x + 15

**Answer:**

Let f(x) = 2x^{4} + 9x^{3} + 2x^{2} + 10x + 15

By factor theorem,

x + 1 = 0 ;x = – 1

If f(– 1) = 0 then (x + 1) is a factor of f(x)

∴f(– 1) = 2(– 1)^{4} + 9(– 1)^{3} + 2(– 1)^{2} + 10(– 1) + 15

= 2 – 9 + 2 – 10 + 15 = 19 – 19 = 0

∴(x + 1) is a factor of f(x) = 2x^{4} + 9x^{3} + 2x^{2} + 10x + 15

**Question 3.**

Determine whether (x + 1) is a factor of the following polynomials:

3x^{3} + 8x^{2} + 6x – 5

**Answer:**

Let f(x) = 3x^{3} + 8x^{2} + 6x – 5

By factor theorem,

x + 1 = 0 ;x = – 1

If f(– 1) = 0 then (x + 1) is a factor of f(x)

∴f(– 1) = 3(– 1)^{3} + 8(– 1)^{2} – 6(– 1) – 5

= – 3 + 8 + 6 – 5 = 6(not equal to 0)

∴(x + 1) is not a factor of f(x) = 3x^{3} + 8x^{2} + 6x – 5

**Question 4.**

Determine whether (x + 1) is a factor of the following polynomials:

x^{3} – 14x^{2} + 3x + 12

**Answer:**

Let f(x) = x^{3} – 14x^{2} + 3x + 12

By factor theorem,

x + 1 = 0 ;x = – 1

If f(– 1) = 0 then (x + 1) is a factor of f(x)

∴f(– 1) = (– 1)^{3} – 14(– 1)^{2} + 3(– 1) + 12

= – 1 – 14 – 3 + 12 = – 6(not equal to 0)

∴(x + 1) is not a factor of f(x) = x^{3} – 14x^{2} + 3x + 12

**Question 5.**

Determine whether (x + 4) is a factor of x^{3} + 3x^{2} – 5x + 36.

**Answer:**

Let f(x) = x^{3} + 3x^{2} – 5x + 36.

By factor theorem,

x + 4 = 0: x = – 4

If f(– 4) = 0, then (x + 4) is a factor

∴f(– 4) = (– 4)^{3} + 3(– 4)^{2} – 5(– 4) + 36

= – 64 + 48 + 20 + 36

= – 64 + 104 = 40

∴f(– 4) is not equal to 0

So, (x + 4) is not a factor of f(x).

**Question 6.**

Using factor theorem show that (x – 1) is a factor of 4x^{3} – 6x^{2} + 9x – 7.

**Answer:**

f(x) = 4x^{3} – 6x^{2} + 9x – 7

By factor theorem,

(x – 1) = 0 ; x = 1

Since, (x – 1) is a factor of f(x)

Therefore, f(1) = 0

f(1) = 4(1)^{3} – 6(1)^{2} + 9(1) – 7 = 4 – 6 + 9 – 7 = 13 – 13 = 0

∴(x – 1) is a factor of f(x)

**Question 7.**

Determine whether (2x + 1) is a factor of 4x^{3} + 4x^{2} – x – 1.

**Answer:**

Let f(x) = 4x^{3} + 4x^{2} – x – 1

By factor Theorem,

2x + 1 = 0 ; x = – 1/2

∴f(– 1/2) = 4(– 1/2)^{3} + 4(– 1/2)^{2} – (– 1/2) – 1

= 4(– 1/8) + 4(1/4) + (1/2) – 1

= (– 1/2) + 1 + (1/2) – 1 = 0

∴f(– 1/2) = 0

So, (2x + 1) is a factor of f(x).

**Question 8.**

Determine the value of p if (x + 3) is a factor of x^{3} – 3x^{2} – px + 24.

**Answer:**

Let f(x) = x^{3} + 3x^{2} – px + 24.

By factor theorem,

x + 3 = 0;x = – 3

∴(x + 3) is a factor of f(x)

So, f(– 3) = 0.

f(– 3) = (– 3)^{3} – 3(– 3)^{2} – p(– 3) + 24 = 0

= >27 – 27 + 3p + 24 = 0

= > – 59 + 24 + 3p = 0

∴3p – 30 = 0

⇒ p = 30/3

**⇒****p = 10**

###### Exercise 3.6

**Question 1.**The coefficient of x^{2} & x in 2x^{3} – 3x^{2} – 2x + 3 are respectively:

A. 2, 3

B. – 3, – 2

C. – 2, – 3

D. 2, – 3

**Answer:**

2x^{3} – 3x^{2} – 2x + 3: Coefficient of x^{2} = – 3

Coefficient of x = – 2

**Question 2.**The degree of polynomial 4x^{2} – 7x^{3} + 6x + 1 is:

A. 2

B. 1

C. 3

D. 0

**Answer:**Degree of polynomial = Highest power of x in the polynomial = 3

**Question 3.**The polynomial 3x – 2 is a :

A. Linear polynomial

B. Quadratic polynomial

C. Cubic polynomial

D. constant polynomial

**Answer:**Given polynomial has degree = 1

**Question 4.**The polynomial 4x^{2} + 2x – 2 is a :

A. Linear polynomial

B. Quadratic polynomial

C. Cubic polynomial

D. constant polynomial

**Answer:**Given polynomial has degree = 2

**Question 5.**The zero of the polynomial 2x – 5:

A. 5/2

B. – 5/2

C. 2/5

D. – 2/5

**Answer:**Given : 2x – 5 = 0

∴ x = 5/2

So, zero of polynomial = 5/2

**Question 6.**The root of polynomial equation 3x – 1 is:

A. – 1/3

B. 1/3

C. 1

D. 3

**Answer:**Given polynomial equation: 3x – 1 = 0

∴ x = 1/3

So, root = 1/3

**Question 7.**The root of polynomial equation x^{2} + 2x = 0:

A. 0, 2

B. 1, 2

C. 1, – 2

D. 0, – 2

**Answer:**Given polynomial equation :

x^{2} + 2x = 0

∴x(x + 2) = 0

x = 0, x + 2 = 0

x = 0, x = – 2

So, the roots are 0 and – 2

**Question 8.**If a polynomial p(x) is divided by (ax + b), then the remainder is:

A. p(b/a)

B. p(– b/a)

C. p(a/b)

D. p(– a/b)

**Answer:**f(x) = ax + b

Therefore, by Remainder Theorem, f(x) = 0

ax + b = 0

x = – b/a ∴Remainder = p(x) = p(– b/a)

**Question 9.**If a polynomial x^{3} – ax^{2} + ax – a is divided by (x – a), then the remainder is:

A. a^{3}

B. a^{2}

C. a

D. – a

**Answer:**Given f(x) = x^{3} – ax^{2} + ax – 2

By Remainder Theorem,

x – a = 0

x = a

∴Remainder = f(a) = a^{3} – a^{3} + 2a – a = a

**Question 10.**If (ax – b) is a factor of p(x) then,

A. p(b) = 0

B. p(– b/a) = 0

C. p(a) = 0

D. p(b/a) = 0

**Answer:**As (ax – b) is a factor of p(x)

ax – b = 0

∴x = b/a So, p(x) = 0 ;

p(b/a) = 0

**Question 11.**One of the factor of x^{2} – 3x – 10 is :

A. x – 2

B. x + 5

C. x – 5

D. x – 3

**Answer:**x^{2} – 3x – 10 = 0

x^{2} – 5x + 2x – 10 = 0

x(x – 5) + 2(x – 5) = 0

(x – 5)(x + 2) = 0 Hence, (x – 5) is a factor.

**Question 12.**One of the factor of x^{3} – 2x^{2} + 2x – 1 is :

A. x – 1

B. x + 1

C. x – 2

D. x + 2

**Answer:**Given: f(x) = x^{3} – 2x^{2} + 2x – 1 = 0

By hit and trial method,

put x = 1

f(1) = 1 – 2 + 2 – 1 = 0

∴(x – 1) is a factor.

**Question 1.**

The coefficient of x^{2} & x in 2x^{3} – 3x^{2} – 2x + 3 are respectively:

A. 2, 3

B. – 3, – 2

C. – 2, – 3

D. 2, – 3

**Answer:**

2x^{3} – 3x^{2} – 2x + 3: Coefficient of x^{2} = – 3

Coefficient of x = – 2

**Question 2.**

The degree of polynomial 4x^{2} – 7x^{3} + 6x + 1 is:

A. 2

B. 1

C. 3

D. 0

**Answer:**

Degree of polynomial = Highest power of x in the polynomial = 3

**Question 3.**

The polynomial 3x – 2 is a :

A. Linear polynomial

B. Quadratic polynomial

C. Cubic polynomial

D. constant polynomial

**Answer:**

Given polynomial has degree = 1

**Question 4.**

The polynomial 4x^{2} + 2x – 2 is a :

A. Linear polynomial

B. Quadratic polynomial

C. Cubic polynomial

D. constant polynomial

**Answer:**

Given polynomial has degree = 2

**Question 5.**

The zero of the polynomial 2x – 5:

A. 5/2

B. – 5/2

C. 2/5

D. – 2/5

**Answer:**

Given : 2x – 5 = 0

∴ x = 5/2

So, zero of polynomial = 5/2

**Question 6.**

The root of polynomial equation 3x – 1 is:

A. – 1/3

B. 1/3

C. 1

D. 3

**Answer:**

Given polynomial equation: 3x – 1 = 0

∴ x = 1/3

So, root = 1/3

**Question 7.**

The root of polynomial equation x^{2} + 2x = 0:

A. 0, 2

B. 1, 2

C. 1, – 2

D. 0, – 2

**Answer:**

Given polynomial equation :

x^{2} + 2x = 0

∴x(x + 2) = 0

x = 0, x + 2 = 0

x = 0, x = – 2

So, the roots are 0 and – 2

**Question 8.**

If a polynomial p(x) is divided by (ax + b), then the remainder is:

A. p(b/a)

B. p(– b/a)

C. p(a/b)

D. p(– a/b)

**Answer:**

f(x) = ax + b

Therefore, by Remainder Theorem, f(x) = 0

ax + b = 0

x = – b/a ∴Remainder = p(x) = p(– b/a)

**Question 9.**

If a polynomial x^{3} – ax^{2} + ax – a is divided by (x – a), then the remainder is:

A. a^{3}

B. a^{2}

C. a

D. – a

**Answer:**

Given f(x) = x^{3} – ax^{2} + ax – 2

By Remainder Theorem,

x – a = 0

x = a

∴Remainder = f(a) = a^{3} – a^{3} + 2a – a = a

**Question 10.**

If (ax – b) is a factor of p(x) then,

A. p(b) = 0

B. p(– b/a) = 0

C. p(a) = 0

D. p(b/a) = 0

**Answer:**

As (ax – b) is a factor of p(x)

ax – b = 0

∴x = b/a So, p(x) = 0 ;

p(b/a) = 0

**Question 11.**

One of the factor of x^{2} – 3x – 10 is :

A. x – 2

B. x + 5

C. x – 5

D. x – 3

**Answer:**

x^{2} – 3x – 10 = 0

x^{2} – 5x + 2x – 10 = 0

x(x – 5) + 2(x – 5) = 0

(x – 5)(x + 2) = 0 Hence, (x – 5) is a factor.

**Question 12.**

One of the factor of x^{3} – 2x^{2} + 2x – 1 is :

A. x – 1

B. x + 1

C. x – 2

D. x + 2

**Answer:**

Given: f(x) = x^{3} – 2x^{2} + 2x – 1 = 0

By hit and trial method,

put x = 1

f(1) = 1 – 2 + 2 – 1 = 0

∴(x – 1) is a factor.