#### 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

#### RD Sharma Mathematics Class 9 by R D Sharma (2018-19 Session)

## Chapter 6: Factorisation of Polynomials

#### Chapter 6: Factorisation of Polynomials Exercise 6.1 solutions [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.

#### Chapter 6: Factorisation of Polynomials Exercise 6.2 solutions [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.

#### Chapter 6: Factorisation of Polynomials Exercise 6.3 solutions [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.

#### Chapter 6: Factorisation of Polynomials Exercise 6.4 solutions [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?

#### Chapter 6: Factorisation of Polynomials Exercise 6.5 solutions [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

#### Chapter 6: Factorisation of Polynomials Exercise 6.5, 0 solutions [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

#### Chapter 6: Factorisation of Polynomials Exercise 0 solutions [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 Mathematics Class 9 by R D Sharma (2018-19 Session)

#### Textbook solutions for Class 9

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

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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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