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# RD Sharma solutions for Class 9 Mathematics chapter 6 - Factorisation of Polynomials

## Chapter 6: Factorisation of Polynomials

Ex. 0Ex. 6.1Ex. 6.2Ex. 6.3Ex. 6.4Ex. 6.5

#### Chapter 6: Factorisation of Polynomials Exercise 6.1 solutions [Pages 2 - 3]

Ex. 6.1 | Q 1.1 | Page 2

Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer:

3x2 - 4x +15

Ex. 6.1 | Q 1.2 | Page 2

Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer:

y^2 +2sqrt3

Ex. 6.1 | Q 1.3 | Page 2

Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer:

3sqrtx+sqrt2x

Ex. 6.1 | Q 1.4 | Page 2

Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer:

x - 4/x

Ex. 6.1 | Q 1.5 | Page 2

Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer:

x^12+y^3+t^50

Ex. 6.1 | Q 2.1 | Page 2

Write the coefficient of x2 in the following:

17 -2x + 7x^2

Ex. 6.1 | Q 2.2 | Page 2

Write the coefficient of x2 in the following:

9-12x +X^3

Ex. 6.1 | Q 2.3 | Page 2

Write the coefficient of x2 in the following:

pi/6x^2- 3x+4

Ex. 6.1 | Q 2.4 | Page 2

Write the coefficient of x2 in the following:

sqrt3x-7

Ex. 6.1 | Q 3.1 | Page 3

Write the degrees of each of the following polynomials

7x3 + 4x2 – 3x + 12

Ex. 6.1 | Q 3.2 | Page 3

Write the degrees of the following polynomials:

12-x+2x^3

Ex. 6.1 | Q 3.3 | Page 3

Write the degrees of the following polynomials:

5y-sqrt2

Ex. 6.1 | Q 3.4 | Page 3

Write the degrees of the following polynomials:

7

Ex. 6.1 | Q 3.5 | Page 3

Write the degrees of the following polynomials

0

Ex. 6.1 | Q 4.1 | Page 3

x+x^2 +4

Ex. 6.1 | Q 4.2 | Page 3

3x-2

Ex. 6.1 | Q 4.3 | Page 3

2x+x^2

Ex. 6.1 | Q 4.4 | Page 3

3y

Ex. 6.1 | Q 4.5 | Page 3

t^2+1

Ex. 6.1 | Q 4.6 | Page 3

7t^4+4t^3+3t-2

Ex. 6.1 | Q 5.1 | Page 3

Classify the following polynomials as polynomials in one-variable, two variables etc:

x^2-xy+7y^2

Ex. 6.1 | Q 5.2 | Page 3

Classify the following polynomials as polynomials in one-variable, two variables etc:

x^2-2tx+7t^2-x+t

Ex. 6.1 | Q 5.3 | Page 3

Classify the following polynomials as polynomials in one-variable, two variables etc:

t^3_3t^2+4t-5

Ex. 6.1 | Q 5.4 | Page 3

Classify the following polynomials as polynomials in one-variable, two variables etc:

xy+yx+zx

Ex. 6.1 | Q 6.1 | Page 3

Identify polynomials in the following:

f(x)=4x^3-x^2-3x+7

Ex. 6.1 | Q 6.2 | Page 3

Identify polynomials in the following:

g(x)=2x^3-3x^2+sqrtx-1

Ex. 6.1 | Q 6.3 | Page 3

Identify polynomials in the following:

p(x)=2/3x^3-7/4x+9

Ex. 6.1 | Q 6.4 | Page 3

Identify polynomials in the following:

q(x)=2x^2-3x+4/x+2

Ex. 6.1 | Q 6.5 | Page 3

Identify polynomials in the following:

h(x)=x^4-x^(3/2)+x-1

Ex. 6.1 | Q 6.6 | Page 3

Identify polynomials in the following:

f(x)=2+3/x+4x

Ex. 6.1 | Q 7.1 | Page 3

Identify constant, linear, quadratic and cubic polynomials from the following polynomials:

f(x)=0

Ex. 6.1 | Q 7.2 | Page 3

Identify constant, linear, quadratic and cubic polynomials from the following polynomials:

g(x)=2x^3-7x+4

Ex. 6.1 | Q 7.3 | Page 3

Identify constant, linear, quadratic and cubic polynomials from the following polynomials:

h(x)=-3x+1/2

Ex. 6.1 | Q 7.4 | Page 3

Identify constant, linear, quadratic and cubic polynomials from the following polynomials

p(x)=2x^2-x+4

Ex. 6.1 | Q 7.5 | Page 3

Identify constant, linear, quadratic and cubic polynomials from the following polynomials:

q(x)=4x+3

Ex. 6.1 | Q 7.6 | Page 3

Identify constant, linear, quadratic and cubic polynomials from the following polynomials:

r(x)=3x^2+4x^2+5x-7

Ex. 6.1 | Q 8 | Page 3

Give one example each of a binomial of degree 35, and of a monomial of degree 100.

#### Chapter 6: Factorisation of Polynomials Exercise 6.2 solutions [Page 8]

Ex. 6.2 | Q 1.1 | Page 8

If f(x)=2x^2-13x^2+17x+12 find f(2)

Ex. 6.2 | Q 1.2 | Page 8

If f(x)=2x^2-13x^2+17x+12 find f-(3)

Ex. 6.2 | Q 1.3 | Page 8

If f(x)=2x^2-13x^2+17x+12 find f(0)

Ex. 6.2 | Q 2.1 | Page 8

Verify whether the indicated numbers is zeroes of the polynomials corresponding to them in the following case:

f ( x ) = 3x +1, x = - 1/3

Ex. 6.2 | Q 2.2 | Page 8

Verify whether the indicated numbers is zeroes of the polynomials corresponding to them in the following case:

f(x)=x^2- 1,x=1,-1

Ex. 6.2 | Q 2.3 | Page 8

Verify whether the indicated numbers is zeroes of the polynomials corresponding to them in the following case:

g(x)=3x^2-2,  x=2/sqrt3     2/sqrt3

Ex. 6.2 | Q 2.4 | Page 8

Verify whether the indicated numbers is zeros of the polynomials corresponding to them in the following case:

$p(x) = x^3 - 6 x^2 + 11x - 6, x = 1, 2, 3$

Ex. 6.2 | Q 2.5 | Page 8

Verify whether the indicated numbers is zeroes of the polynomials corresponding to them in the following case:

f ( x ) = 5x - pi , x = 4/5

Ex. 6.2 | Q 2.6 | Page 8

Verify whether the indicated numbers is zeroes of the polynomials corresponding to them in the following case:

f ( x) = x^2and x = 0

Ex. 6.2 | Q 2.7 | Page 8

Verify whether the indicated numbers is zeroes of the polynomials corresponding to them in the following case:

f(x) = lx + m , x = - m/1

Ex. 6.2 | Q 2.8 | Page 8

Verify whether the indicated numbers is zeroes of the polynomials corresponding to them in the following case:

f (x) = 2x +1, x = 1/2

Ex. 6.2 | Q 3 | Page 8

If x = 2 is a root of the polynomial f(x) = 2x2 – 3x + 7a find the value of a.

Ex. 6.2 | Q 4 | Page 8

If x = −1/2 is a zero of the polynomial p(x)=8x^3-ax^2 -+2 find the value of a.

Ex. 6.2 | Q 5 | Page 8

If x = 0 and x = −1 are the roots of the polynomial f(x) =2x3 − 3x2 + ax + b, find the value of a and b.

Ex. 6.2 | Q 6 | Page 8

Find the integral roots of the polynomial f(x) = x3 + 6x2 + 11x + 6.

Ex. 6.2 | Q 7 | Page 8

Find rational roots of the polynomial f(x) = 2x3 + x2 − 7x − 6.

#### Chapter 6: Factorisation of Polynomials Exercise 6.3 solutions [Pages 14 - 15]

Ex. 6.3 | Q 1 | Page 14

In each of the following, using the remainder theorem, find the remainder when f(x) is divided by g(x) and verify the result by actual division: (1−8)

f(x) = x3 + 4x2 − 3x + 10, g(x) = x + 4

Ex. 6.3 | Q 2 | Page 14

f(x) = 4x4 − 3x3 − 2x2 + x − 7, g(x) = x − 1

Ex. 6.3 | Q 3 | Page 14

f(x) = 2x4 − 6x3 + 2x2 − x + 2, g(x) = x + 2

Ex. 6.3 | Q 4 | Page 14

f(x) = 4x3 − 12x2 + 14x − 3, g(x) 2x − 1

Ex. 6.3 | Q 5 | Page 14

f(x) = x3 − 6x2 + 2x − 4, g(x) = 1 − 2x

Ex. 6.3 | Q 6 | Page 14

f(x) = x4 − 3x2 + 4, g(x) = x − 2

Ex. 6.3 | Q 7 | Page 14

f(x) = 9x3 − 3x2 + x − 5, g(x) = $x - \frac{2}{3}$

Ex. 6.3 | Q 8 | Page 14

$f(x) = 3 x^4 + 2 x^3 - \frac{x^2}{3} - \frac{x}{9} + \frac{2}{27}, g(x) = x + \frac{2}{3}$

Ex. 6.3 | Q 9 | Page 14

If the polynomials 2x3 + ax2 + 3x − 5 and x3 + x2 − 4x +a leave the same remainder when divided by x −2, find the value of a.

Ex. 6.3 | Q 10 | Page 14

If the polynomials ax3 + 3x2 − 13 and 2x3 − 5x + a, when divided by (x − 2) leave the same remainder, find the value of a.

Ex. 6.3 | Q 11.1 | Page 14

Find the remainder when x3 + 3x2 + 3x + 1 is divided by  x + 1.

Ex. 6.3 | Q 11.2 | Page 14

Find the remainder when x3 + 3x2 + 3x + 1 is divided by $x - \frac{1}{2}$.

Ex. 6.3 | Q 11.3 | Page 14

Find the remainder when x3 + 3x2 + 3x + 1 is divided by x.

Ex. 6.3 | Q 11.4 | Page 14

Find the remainder when x3 + 3x2 + 3x + 1 is divided by $x + \pi$ .

Ex. 6.3 | Q 11.5 | Page 14

Find the remainder when x3 + 3x2 + 3x + 1 is divided by 5 + 2x .

Ex. 6.3 | Q 12.1 | Page 15

The polynomials ax3 + 3x2 − 3 and 2x3 − 5x + a when divided by (x − 4) leave the remainders R1 and R2 respectively. Find the value of the following case, if R1 = R2.

Ex. 6.3 | Q 12.2 | Page 15

The polynomials ax3 + 3x2 − 3 and 2x3 − 5x + a when divided by (x − 4) leave the remainders R1 and R2 respectively. Find the values of the following case, if  R1 + R2 = 0.

Ex. 6.3 | Q 12.3 | Page 15

The polynomials ax3 + 3x2 − 3 and 2x3 − 5x + a when divided by (x − 4) leave the remainders R1 and R2 respectively. Find the values of the following cases, if 2R1 − R2 = 0.

#### Chapter 6: Factorisation of Polynomials Exercise 6.4 solutions [Pages 24 - 25]

Ex. 6.4 | Q 1 | Page 24

In each of the following, use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x) or, not: (1−7)

f(x) = x3 − 6x2 + 11x − 6; g(x) = x − 3

Ex. 6.4 | Q 2 | Page 24

f(x) = 3x4 + 17x3 + 9x2 − 7x − 10; g(x) = x + 5

Ex. 6.4 | Q 3 | Page 24

f(x) = x5 + 3x4 − x3 − 3x2 + 5x + 15, g(x) = x + 3

Ex. 6.4 | Q 4 | Page 24

f(x) = x3 −6x2 − 19x + 84, g(x) = x − 7

Ex. 6.4 | Q 5 | Page 24

f(x) = 3x3 + x2 − 20x +12, g(x) = 3x − 2

Ex. 6.4 | Q 6 | Page 24

f(x) = 2x3 − 9x2 + x + 12, g(x) = 3 − 2x

Ex. 6.4 | Q 7 | Page 24

f(x) = x3 − 6x2 + 11x − 6, g(x) = x2 − 3x + 2

Ex. 6.4 | Q 8 | Page 24

Show that (x − 2), (x + 3) and (x − 4) are factors of x3 − 3x2 − 10x + 24.

Ex. 6.4 | Q 9 | Page 24

Show that (x + 4) , (x − 3) and (x − 7) are factors of x3 − 6x2 − 19x + 84

Ex. 6.4 | Q 10 | Page 24

For what value of a is (x − 5) a factor of x3 − 3x2 + ax − 10?

Ex. 6.4 | Q 11 | Page 24

Find the value of a such that (x − 4)  is a factors of 5x3 − 7x2 − ax − 28.

Ex. 6.4 | Q 12 | Page 24

Find the value of a, if x + 2 is a factor of 4x4 + 2x3 − 3x2 + 8x + 5a.

Ex. 6.4 | Q 13 | Page 24

Find the value k if x − 3 is a factor of k2x3 − kx2 + 3kx − k.

Ex. 6.4 | Q 14 | Page 24

Find the values of a and b, if x2 − 4 is a factor of ax4 + 2x3 − 3x2 + bx − 4

Ex. 6.4 | Q 15 | Page 24

Find α and β, if x + 1 and x + 2 are factors of x3 + 3x2 − 2αx + β.

Ex. 6.4 | Q 16.1 | Page 24

If x − 2 is a factor of the following two polynomials, find the values of a in each case x3 − 2ax2 + ax − 1.

Ex. 6.4 | Q 16.2 | Page 24

If x − 2 is a factor of the following two polynomials, find the values of a in each case x5 − 3x4 − ax3 + 3ax2 + 2ax + 4.

Ex. 6.4 | Q 17.1 | Page 25

In the following two polynomials, find the value of a, if x − a is  factor x6 − ax5 + x4 − ax3 + 3x − a + 2.

Ex. 6.4 | Q 17.2 | Page 25

In the following two polynomials, find the value of a, if x − a is  factor  (x5 − a2x3 + 2x + a + 1).

Ex. 6.4 | Q 18.1 | Page 25

In the following two polynomials, find the value of a, if x + a is a factor x3 + ax2 − 2x +a + 4.

Ex. 6.4 | Q 18.2 | Page 25

In the following two polynomials, find the value of a, if x + a is a factor x4 − a2x2 + 3x −a.

Ex. 6.4 | Q 19 | Page 25

Find the values of p and q so that x4 + px3 + 2x3 − 3x + q is divisible by (x2 − 1).

Ex. 6.4 | Q 20 | Page 25

Find the values of a and b so that (x + 1) and (x − 1) are factors of x4 + ax3 − 3x2 + 2x + b.

Ex. 6.4 | Q 21 | Page 25

If x3 + ax2 − bx+ 10 is divisible by x2 − 3x + 2, find the values of a and b

Ex. 6.4 | Q 22 | Page 25

If both x + 1 and x − 1 are factors of ax3 + x2 − 2x + b, find the values of a and b.

Ex. 6.4 | Q 23 | Page 25

What must be added to x3 − 3x2 − 12x + 19 so that the result is exactly divisibly by x2 + x - 6 ?

Ex. 6.4 | Q 24 | Page 25

What must be subtracted from x3 − 6x2 − 15x + 80 so that the result is exactly divisible by x2 + x − 12?

Ex. 6.4 | Q 25 | Page 25

What must be added to 3x3 + x2 − 22x + 9 so that the result is exactly divisible by 3x2 + 7x − 6?

#### Chapter 6: Factorisation of Polynomials Exercise 6.5 solutions [Pages 32 - 33]

Ex. 6.5 | Q 1 | Page 32

Using factor theorem, factorize each of the following polynomials:
x3 + 6x2 + 11x + 6

Ex. 6.5 | Q 2 | Page 32

x3 + 2x2 − x − 2

Ex. 6.5 | Q 3 | Page 32

x3 − 6x2 + 3x + 10

Ex. 6.5 | Q 4 | Page 32

x4 − 7x3 + 9x2 + 7x − 10

Ex. 6.5 | Q 5 | Page 32

3x3 − x2 − 3x + 1

Ex. 6.5 | Q 6 | Page 32

x3 − 23x2 + 142x − 120

Ex. 6.5 | Q 7 | Page 33

y3 − 7y + 6

Ex. 6.5 | Q 8 | Page 33

x3 − 10x2 − 53x − 42

Ex. 6.5 | Q 9 | Page 33

y3 − 2y2 − 29y − 42

Ex. 6.5 | Q 10 | Page 33

2y3 − 5y2 − 19y + 42

Ex. 6.5 | Q 11 | Page 33

x3 + 13x2 + 32x + 20

Ex. 6.5 | Q 12 | Page 33

x3 − 3x2 − 9x − 5

Ex. 6.5 | Q 13 | Page 33

2y3 + y2 − 2y − 1

Ex. 6.5 | Q 14 | Page 33

x3 − 2x2 − x + 2

Ex. 6.5 | Q 15.1 | Page 33

Factorize of the following polynomials:

x3 + 13x2 + 31x − 45 given that x + 9 is a factor

Ex. 6.5 | Q 15.2 | Page 33

Factorize of the following polynomials:

4x3 + 20x2 + 33x + 18 given that 2x + 3 is a factor.

Ex. 6.5 | Q 16 | Page 33

x4 − 2x3 − 7x2 + 8x + 12

Ex. 6.5 | Q 17 | Page 33

x4 + 10x3 + 35x2 + 50x + 24

Ex. 6.5 | Q 18 | Page 33

2x4 − 7x3 − 13x2 + 63x − 45

#### Chapter 6: Factorisation of Polynomials Exercise 6.5, 0 solutions [Pages 33 - 34]

Ex. 6.5 | Q 1 | Page 33

Define zero or root of a polynomial.

Ex. 6.5 | Q 2 | Page 33

If $x = \frac{1}{2}$ is a zero of the polynomial f(x) = 8x3 + ax2 − 4x + 2, find the value of a.

Ex. 6.5 | Q 3 | Page 33

Write the remainder when the polynomialf(x) = x3 + x2 − 3x + 2 is divided by x + 1.

Ex. 6.5 | Q 4 | Page 33

Find the remainder when x3 + 4x2 + 4x − 3  is divided by x.

Ex. 6.5 | Q 5 | Page 33

If x + 1 is a factor of x3 + a, then write the value of a.

Ex. 0 | Q 6 | Page 34

If f(x) = x4 − 2x3 + 3x2 − ax − b when divided by x − 1, the remainder is 6, then find the value of a + b

#### Chapter 6: Factorisation of Polynomials Exercise 0 solutions [Pages 34 - 35]

Ex. 0 | Q 1 | Page 34

Mark the correct alternative in each of the following:
If x − 2 is a factor of x2 + 3ax − 2a, then a =

• 2

• -2

• 1

• -1

Ex. 0 | Q 2 | Page 34

If x3 + 6x2 + 4x + k is exactly divisible by x + 2, then k =

• −6

• −7

• −8

• −10

Ex. 0 | Q 3 | Page 34

If x − a is a factor of x3 −3x2a + 2a2x + b, then the value of b is

• 0

• 2

• 1

• 3

Ex. 0 | Q 4 | Page 34

If x140 + 2x151 + k is divisible by x + 1, then the value of k is

• 1

• -3

• 2

• -2

Ex. 0 | Q 5 | Page 34

If x + 2 is a factor of x2 + mx + 14, then m =

• 7

• 2

• 9

• 14

Ex. 0 | Q 6 | Page 34

If x − 3 is a factor of x2 − ax − 15, then a =

• -2

• 5

• -5

• 3

Ex. 0 | Q 7 | Page 34

If x51 + 51 is divided by x + 1, the remainder is

• 0

• 1

• 49

• 50

Ex. 0 | Q 8 | Page 34

If x + 1 is a factor of the polynomial 2x2 + kx, then k =

• -2

• -3

• 4

• 2

Ex. 0 | Q 9 | Page 34

If x + a is a factor of x4 − a2x2 + 3x − 6a, then a =

• 0

• -1

• 1

• 2

Ex. 0 | Q 10 | Page 34

The value of k for which x − 1 is a factor of 4x3 + 3x2 − 4x + k, is

• 3

• 1

• -2

• -3

Ex. 0 | Q 11 | Page 34

If x + 2  and x − 1 are the factors of x3 + 10x2 + mx + n, then the values of m and n are respectively

• 5 and −3

• 17 and −8

• 7 and −18

•  23 and −19

Ex. 0 | Q 12 | Page 34

Let f(x)  be a polynomial such that $f\left( - \frac{1}{2} \right)$ = 0, then a factor of f(x) is

• 2x − 1

• 2x + 1

• x − 1

•  x +1

Ex. 0 | Q 13 | Page 35

When x3 − 2x2 + ax − b is divided by x2 − 2x − 3, the remainder is x − 6. The values of a and b are respectively

• −2, −6

•  2 and −6

• - 2 and 6

•  2 and 6

Ex. 0 | Q 14 | Page 35

One factor of x4 + x2 − 20 is x2 + 5. The other factor is

•  x2 − 4

•  x − 4

• x2 − 5

• x + 2

Ex. 0 | Q 15 | Page 35

If (x − 1) is a factor of polynomial f(x) but not of g(x) , then it must be a factor of

• f(x) g(x)

•  −f(x) + g(x)

• f(x) − g(x)

• $\left\{ f(x) + g(x) \right\} g(x)$

Ex. 0 | Q 16 | Page 35

(x+1) is a factor of xn + 1 only if

• n is an odd integer

• n is an even integer

• n is a negative integer

• n is a positive integer

Ex. 0 | Q 17 | Page 35

If x2 + x + 1 is a factor of the polynomial 3x3 + 8x2 + 8x + 3 + 5k, then the value of k is

• 0

• 2/5

• 5/2

• -1

Ex. 0 | Q 18 | Page 35

If (3x − 1)7  = a7x7 + a6x6 + a5x5 +...+ a1x + a0, then a7 + a5 + ...+a1 + a=

• 0

• 1

• 128

• 64

Ex. 0 | Q 19 | Page 35

If both x − 2 and $x - \frac{1}{2}$ are factors of px2 + 5x + r, then

•  p = r

• p + r = 0

• 2p + r = 0

• p + 2r = 0

Ex. 0 | Q 20 | Page 35

If x2 − 1 is a factor of ax4 + bx3 + cx2 + dx + e, then

• a + c + e = b + d

• a + b +e = c + d

• a + b + c = d + e

• b + c + d = a + e

## Chapter 6: Factorisation of Polynomials

Ex. 0Ex. 6.1Ex. 6.2Ex. 6.3Ex. 6.4Ex. 6.5

## RD Sharma solutions for Class 9 Mathematics chapter 6 - Factorisation of Polynomials

RD Sharma solutions for Class 9 Maths chapter 6 (Factorisation of Polynomials) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics for Class 9 by R D Sharma (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 9 Mathematics chapter 6 Factorisation of Polynomials are Introduction of Polynomials, Polynomials in One Variable, Zeroes of a Polynomial, Remainder Theorem, Factorisation of Polynomials.

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