#### Chapters

Chapter 2: Exponents of Real Numbers

Chapter 3: Rationalisation

Chapter 4: Algebraic Identities

Chapter 5: Factorisation of Algebraic Expressions

Chapter 6: Factorisation of Polynomials

Chapter 7: Linear Equations in Two Variables

Chapter 8: Co-ordinate Geometry

Chapter 9: Introduction to Euclid’s Geometry

Chapter 10: Lines and Angles

Chapter 11: Triangle and its Angles

Chapter 12: Congruent Triangles

Chapter 13: Quadrilaterals

Chapter 14: Areas of Parallelograms and Triangles

Chapter 15: Circles

Chapter 16: Constructions

Chapter 17: Heron’s Formula

Chapter 18: Surface Areas and Volume of a Cuboid and Cube

Chapter 19: Surface Areas and Volume of a Circular Cylinder

Chapter 20: Surface Areas and Volume of A Right Circular Cone

Chapter 21: Surface Areas and Volume of a Sphere

Chapter 22: Tabular Representation of Statistical Data

Chapter 23: Graphical Representation of Statistical Data

Chapter 24: Measures of Central Tendency

Chapter 25: Probability

## Chapter 6: Factorisation of Polynomials

#### Exercise 6.1 [Pages 2 - 3]

### RD Sharma solutions for Mathematics for Class 9 Chapter 6 Factorisation of Polynomials Exercise 6.1 [Pages 2 - 3]

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

3x^{2} - 4x +15

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

*`y^2 +2sqrt3`*

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

`3sqrtx+sqrt2x`

`x - 4/x`

`x^12+y^3+t^50`

Write the coefficient of x^{2} in the following:

`17 -2x + 7x^2`

Write the coefficient of x^{2} in the following:

`9-12x +X^3`

Write the coefficient of x^{2} in the following:

`pi/6x^2- 3x+4`

Write the coefficient of x^{2} in the following:

`sqrt3x-7`

Write the degrees of each of the following polynomials

`7x3 + 4x2 – 3x + 12`

Write the degrees of the following polynomials:

`12-x+2x^3`

Write the degrees of the following polynomials:

`5y-sqrt2`

Write the degrees of the following polynomials:

7

Write the degrees of the following polynomials

0

Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials:

`x+x^2 +4`

Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials:

`3x-2`

Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials:

`2x+x^2`

Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials:

`3y`

Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials:

`t^2+1`

Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials

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

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

`x^2-xy+7y^2`

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

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

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

`t^3_3t^2+4t-5`

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

`xy+yx+zx`

Identify polynomials in the following:

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

Identify polynomials in the following:

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

Identify polynomials in the following:

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

Identify polynomials in the following:

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

Identify polynomials in the following:

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

Identify polynomials in the following:

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

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

`f(x)=0`

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

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

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

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

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

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

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

`q(x)=4x+3`

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

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

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

#### Exercise 6.2 [Page 8]

### RD Sharma solutions for Mathematics for Class 9 Chapter 6 Factorisation of Polynomials Exercise 6.2 [Page 8]

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

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

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

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

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

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`

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`

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

\[p(x) = x^3 - 6 x^2 + 11x - 6, x = 1, 2, 3\]

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

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

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

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

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

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

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

Find the integral roots of the polynomial f(x) = x^{3} + 6x^{2} + 11x + 6.

Find rational roots of the polynomial f(x) = 2x^{3} + x^{2} − 7x − 6.

#### Exercise 6.3 [Pages 14 - 15]

### RD Sharma solutions for Mathematics for Class 9 Chapter 6 Factorisation of Polynomials Exercise 6.3 [Pages 14 - 15]

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) = x^{3} + 4x^{2} − 3x + 10, g(x) = x + 4

f(x) = 4x^{4} − 3x^{3} − 2x^{2} + x − 7, g(x) = x − 1

f(x) = 2x^{4} − 6x^{3} + 2x^{2} − x + 2, g(x) = x + 2

f(x) = 4x^{3} − 12x^{2} + 14x − 3, g(x) 2x − 1

f(x) = x^{3} − 6x^{2} + 2x − 4, g(x) = 1 − 2x

f(x) = x^{4} − 3x^{2} + 4, g(x) = x − 2

f(x) = 9x^{3} − 3x^{2} + x − 5, g(x) = \[x - \frac{2}{3}\]

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

If the polynomials 2x^{3} + ax^{2} + 3x − 5 and x^{3} + x^{2} − 4x +a leave the same remainder when divided by x −2, find the value of a.

If the polynomials *a*x^{3} + 3x^{2} − 13 and 2x^{3} − 5x + a, when divided by (x − 2) leave the same remainder, find the value of a.

Find the remainder when x^{3} + 3x^{2} + 3x + 1 is divided by x + 1.

Find the remainder when x^{3} + 3x^{2} + 3x + 1 is divided by \[x - \frac{1}{2}\].

Find the remainder when x^{3} + 3x^{2} + 3x + 1 is divided by x.

Find the remainder when x^{3} + 3x^{2} + 3x + 1 is divided by \[x + \pi\] .

Find the remainder when x^{3} + 3x^{2} + 3x + 1 is divided by 5 + 2x .

The polynomials ax^{3} + 3x^{2} − 3 and 2x^{3} − 5x + a when divided by (x − 4) leave the remainders R_{1} and R_{2} respectively. Find the value of the following case, if R_{1} = R_{2.}

The polynomials ax^{3} + 3x^{2} − 3 and 2x^{3} − 5x + a when divided by (x − 4) leave the remainders R_{1} and R_{2} respectively. Find the values of the following case, if R_{1} + R_{2} = 0.

The polynomials ax^{3} + 3x^{2} − 3 and 2x^{3} − 5x + a when divided by (x − 4) leave the remainders R_{1} and R_{2} respectively. Find the values of the following cases, if 2R_{1} − R_{2} = 0.

#### Exercise 6.4 [Pages 24 - 25]

### RD Sharma solutions for Mathematics for Class 9 Chapter 6 Factorisation of Polynomials Exercise 6.4 [Pages 24 - 25]

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) = x^{3} − 6x^{2} + 11x − 6; g(x) = x − 3

f(x) = 3x^{4} + 17x^{3} + 9x^{2} − 7x − 10; g(x) = x + 5

f(x) = x^{5} + 3x^{4} − x^{3} − 3x^{2} + 5x + 15, g(x) = x + 3

f(x) = x^{3} −6x^{2} − 19x + 84, g(x) = x − 7

f(x) = 3x^{3} + x^{2} − 20x +12, g(x) = 3x − 2

f(x) = 2x^{3} − 9x^{2} + x + 12, g(x) = 3 − 2x

f(x) = x^{3} − 6x^{2} + 11x − 6, g(x) = x^{2} − 3x + 2

Show that (x − 2), (x + 3) and (x − 4) are factors of x^{3} − 3x^{2} − 10x + 24.

Show that (x + 4) , (x − 3) and (x − 7) are factors of x^{3} − 6x^{2} − 19x + 84

For what value of a is (x − 5) a factor of x^{3}^{ }− 3x^{2} + ax − 10?

Find the value of *a* such that (x − 4) is a factors of 5x^{3} − 7x^{2} − ax − 28.

Find the value of a, if x + 2 is a factor of 4x^{4} + 2x^{3} − 3x^{2} + 8x + 5a.

Find the value k if x − 3 is a factor of k^{2}x^{3} − kx^{2} + 3kx − k.

Find the values of a and b, if x^{2} − 4 is a factor of ax^{4} + 2x^{3} − 3x^{2} + bx − 4

Find α and β, if x + 1 and x + 2 are factors of x^{3} + 3x^{2}^{ }− 2αx + β.

If x − 2 is a factor of the following two polynomials, find the values of a in each case x^{3} − 2ax^{2} + ax − 1.

If x − 2 is a factor of the following two polynomials, find the values of a in each case x^{5} − 3x^{4} − ax^{3} + 3ax^{2} + 2ax + 4.

In the following two polynomials, find the value of a, if x − a is factor x^{6} − ax^{5} + x^{4} − ax^{3} + 3x − a + 2.

In the following two polynomials, find the value of a, if x − a is factor (x^{5} − a^{2}x^{3} + 2x + a + 1).

In the following two polynomials, find the value of a, if x + a is a factor x^{3} + ax^{2} − 2x +a + 4.

In the following two polynomials, find the value of a, if x + a is a factor x^{4} − a^{2}x^{2} + 3x −a.

Find the values of *p* and *q* so that x^{4} + px^{3} + 2x^{3} − 3x + q is divisible by (x^{2} − 1).

Find the values of a and b so that (x + 1) and (x − 1) are factors of x^{4} + ax^{3} − 3x^{2} + 2x + b.

If x^{3} + ax^{2} − bx+ 10 is divisible by x^{2} − 3x + 2, find the values of a and b

If both x + 1 and x − 1 are factors of ax^{3} + x^{2} − 2x + b, find the values of a and b.

What must be added to x^{3} − 3x^{2} − 12x + 19 so that the result is exactly divisibly by x^{2} + x - 6 ?

What must be subtracted from x^{3} − 6x^{2} − 15x + 80 so that the result is exactly divisible by x^{2} + x − 12?

What must be added to 3x^{3} + x^{2} − 22x + 9 so that the result is exactly divisible by 3x^{2} + 7x − 6?

#### Exercise 6.5 [Pages 32 - 33]

### RD Sharma solutions for Mathematics for Class 9 Chapter 6 Factorisation of Polynomials Exercise 6.5 [Pages 32 - 33]

Using factor theorem, factorize each of the following polynomials:

x^{3} + 6x^{2} + 11x + 6

x^{3} + 2x^{2} − x − 2

x^{3} − 6x^{2} + 3x + 10

x^{4} − 7x^{3}_{ }+ 9x^{2} + 7x − 10

3x^{3} − x^{2} − 3x + 1

x^{3} − 23x^{2} + 142x − 120

y^{3} − 7y + 6

x^{3} − 10x^{2} − 53x − 42

y^{3} − 2y^{2} − 29y − 42

2y^{3} − 5y^{2} − 19y + 42

x^{3} + 13x^{2} + 32x + 20

x^{3} − 3x^{2} − 9x − 5

2y^{3} + y^{2} − 2y − 1

x^{3} − 2x^{2} − x + 2

Factorize of the following polynomials:

x^{3} + 13x^{2} + 31x − 45 given that x + 9 is a factor

Factorize of the following polynomials:

4*x*^{3} + 20*x*^{2} + 33*x* + 18 given that 2*x* + 3 is a factor.

x^{4} − 2x^{3} − 7x^{2} + 8x + 12

x^{4} + 10x^{3} + 35x^{2} + 50x + 24

2x^{4} − 7x^{3} − 13x^{2} + 63x − 45

#### Exercise 6.5 [Pages 33 - 34]

### RD Sharma solutions for Mathematics for Class 9 Chapter 6 Factorisation of Polynomials Exercise 6.5 [Pages 33 - 34]

Define zero or root of a polynomial.

If \[x = \frac{1}{2}\] is a zero of the polynomial f(x) = 8x^{3} + ax^{2} − 4x + 2, find the value of a.

Write the remainder when the polynomialf(x) = x^{3} + x^{2} − 3x + 2 is divided by x + 1.

Find the remainder when* *x^{3} + 4x^{2} + 4x − 3 is divided by x.

If x + 1 is a factor of x^{3} + a, then write the value of a.

If f(x) = x^{4} − 2x^{3} + 3x^{2} − ax − b when divided by x − 1, the remainder is 6, then find the value of a + b

#### [Pages 34 - 35]

### RD Sharma solutions for Mathematics for Class 9 Chapter 6 Factorisation of Polynomials [Pages 34 - 35]

Mark the correct alternative in each of the following:

If x − 2 is a factor of x^{2} + 3ax − 2a, then a =

2

-2

1

-1

If x^{3} + 6x^{2} + 4x + k is exactly divisible by x + 2, then k =

−6

−7

−8

−10

If x − a is a factor of x^{3} −3x^{2}a + 2a^{2}x + b, then the value of b is

0

2

1

3

If x^{140} + 2x^{151} + k is divisible by x + 1, then the value of k is

1

-3

2

-2

If x + 2 is a factor of x^{2} + mx + 14, then m =

7

2

9

14

If x − 3 is a factor of x^{2} − ax − 15, then a =

-2

5

-5

3

If x^{51} + 51 is divided by x + 1, the remainder is

0

1

49

50

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

-2

-3

4

2

If x + a is a factor of x^{4} − a^{2}x^{2} + 3x − 6a, then a =

0

-1

1

_{2}

The value of k for which x − 1 is a factor of 4x^{3} + 3x^{2} − 4x + k, is

3

1

-2

-3

If x + 2 and x − 1 are the factors of x^{3} + 10x^{2} + mx + n, then the values of m and n are respectively

5 and −3

17 and −8

7 and −18

23 and −19

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

When x^{3} − 2x^{2} + ax − b is divided by x^{2} − 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

One factor of x^{4} + x^{2} − 20 is x^{2} + 5. The other factor is

x

^{2}− 4x − 4

x

^{2}− 5x + 2

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)

\[\left\{ f(x) + g(x) \right\} g(x)\]

(x+1) is a factor of x^{n} + 1 only if

n is an odd integer

n is an even integer

n is a negative integer

n is a positive integer

If x^{2} + x + 1 is a factor of the polynomial 3x^{3} + 8x^{2}^{ }+ 8x + 3 + 5k, then the value of k is

0

2/5

5/2

-1

If (3x − 1)^{7} = a_{7}x^{7} + a_{6}x^{6} + a_{5}x^{5} +...+ a_{1}x + a_{0}, then a_{7} + a_{5}_{ }+ ...+a_{1} + a_{0 }=

0

1

128

64

If both x − 2 and \[x - \frac{1}{2}\] are factors of px^{2} + 5x + r, then

p = r

p + r = 0

2p + r = 0

p + 2r = 0

If x^{2} − 1 is a factor of ax^{4} + bx^{3} + cx^{2} + 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

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

RD Sharma solutions for Mathematics for Class 9 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 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Mathematics for Class 9 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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