## RD Sharma Solutions for Class 9 Maths Chapter 6 – Free PDF Download

**RD Sharma Solutions for Class 9 Maths Chapter 6** **Factorization of Polynomials** are provided here to enhance conceptual understanding among students. In this chapter, students will learn various terms such as polynomial, the degree of a polynomial, factors, multiples and zeros of a polynomial. A polynomial is defined as an expression which is composed of variables, constants and exponents that are combined using mathematical operations such as addition, subtraction, multiplication and division (No division operation by a variable). Based on the number of terms present in the expression, it is classified as monomial, binomial and trinomial. Students can download RD Sharma Solutions for Class 9 to learn all chapters listed in the textbook in depth.

PDFs of these solutions can be easily downloaded from the links below. BYJU’S subject-matter experts have uniquely formulated the solutions for a better understanding of the covered concepts. By practising RD Sharma Solutions for Class 9, students can achieve their aim of securing high marks in the final exam. **RD Sharma Solutions** are the best study guide to grasp the concepts better.

## RD Sharma Solutions for Class 9 Maths Chapter 6 Factorization of Polynomials

### Access Answers to Maths RD Sharma Solutions for Class 9 Chapter 6 Factorization of Polynomials

### Exercise 6.1 Page No: 6.2

**Question 1: Which of the following expressions are polynomials in one variable, and which are not? State reasons for your answer:**

**(i) 3x ^{2} – 4x + 15**

**(ii) y ^{2} + 2√3**

**(iii) 3√x + √2x**

**(iv) x – 4/x**

**(v) x ^{12} + y^{3} + t^{50}**

**Solution:**

**(i)** 3x^{2} – 4x + 15

*It is a polynomial of x.*

**(ii)** y^{2} + 2√3

*It is a polynomial of y.*

**(iii)** 3√x + √2x

*It is not a polynomial since the exponent of 3√x is a rational term.*

**(iv)** x – 4/x

*It is not a polynomial since the exponent of – 4/x is not a positive term.*

**(v)** x^{12} + y^{3} + t^{50}

*It is a three-variable polynomial, x, y and t.*

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

**(i) 17 – 2x + 7x ^{2}**

**(ii) 9 – 12x + x ^{3}**

**(iii) ∏/6 x ^{2} – 3x + 4**

**(iv) √3x – 7**

**Solution:**

**(i)** 17 – 2x + 7x^{2}

*Coefficient of x ^{2} = 7*

**(ii)** 9 – 12x + x^{3}

*Coefficient of x ^{2 }=0*

**(iii)** ∏/6 x^{2} – 3x + 4

*Coefficient of x ^{2 }= ∏/6*

**(iv)** √3x – 7

*Coefficient of x ^{2 }= 0*

**Question 3: Write the degrees of each of the following polynomials:**

**(i) 7x ^{3} + 4x^{2} – 3x + 12**

**(ii) 12 – x + 2x ^{3}**

**(iii) 5y – √2**

**(iv) 7**

**(v) 0**

**Solution**:

As we know, degree is the highest power in the polynomial

**(i)** Degree of the polynomial 7x^{3} + 4x^{2} – 3x + 12 is *3*

**(ii)** Degree of the polynomial 12 – x + 2x^{3} is *3*

**(iii)** Degree of the polynomial 5y – √2 is *1*

**(iv)** Degree of the polynomial 7 is *0*

**(v)** Degree of the polynomial 0 is *undefined.*

**Question 4: Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials:**

**(i) x + x ^{2} + 4**

**(ii) 3x – 2**

**(iii) 2x + x ^{2}**

**(iv) 3y**

**(v) t ^{2} + 1**

**(vi) 7t ^{4} + 4t^{3} + 3t – 2**

**Solution:**

**(i)** x + x^{2} + 4: It is a quadratic polynomial as its degree is *2*.

**(ii)** 3x – 2 : It is a linear polynomial as its degree is *1*.

**(iii)** 2x + x^{2}: It is a quadratic polynomial as its degree is *2*.

**(iv)** 3y: It is a linear polynomial as its degree is *1*.

**(v)** t^{2}+ 1: It is a quadratic polynomial as its degree is *2*.

**(vi)** 7t^{4} + 4t^{3} + 3t – 2: It is a biquadratic polynomial as its degree is *4*.

### Exercise 6.2 Page No: 6.8

**Question 1: If f(x) = 2x ^{3} – 13x^{2} + 17x + 12, find**

**(i) f (2)**

**(ii) f (-3)**

**(iii) f(0)**

**Solution:**

f(x) = 2x^{3} – 13x^{2} + 17x + 12

**(i)** f(2) = 2(2)^{3} – 13(2)^{ 2} + 17(2) + 12

= 2 x 8 – 13 x 4 + 17 x 2 + 12

= 16 – 52 + 34 + 12

= 62 – 52

*= 10*

**(ii)** f(-3) = 2(-3)^{3} – 13(-3)^{ 2} + 17 x (-3) + 12

= 2 x (-27) – 13 x 9 + 17 x (-3) + 12

= -54 – 117 -51 + 12

= -222 + 12

*= -210*

**(iii)** f(0) = 2 x (0)^{3} – 13(0)^{ 2} + 17 x 0 + 12

= 0-0 + 0+ 12

*= 12*

**Question 2: Verify whether the indicated numbers are zeros of the polynomials corresponding to them in the following cases:**

**(i) f(x) = 3x + 1, x = −1/3**

**(ii) f(x) = x ^{2} – 1, x = 1,−1**

**(iii) g(x) = 3x ^{2} – 2 , x = 2/√3 , −2/√3 **

**(iv) p(x) = x ^{3} – 6x^{2} + 11x – 6 , x = 1, 2, 3**

**(v) f(x) = 5x – π, x = 4/5**

**(vi) f(x) = x ^{2} , x = 0**

**(vii) f(x) = lx + m, x = −m/l**

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

**Solution:**

**(i)** f(x) = 3x + 1, x = −1/3

f(x) = 3x + 1

Substitute x = −1/3 in f(x)

f( −1/3) = 3(−1/3) + 1

= -1 + 1

= 0

*Since, the result is 0, so x = −1/3 is the root of 3x + 1*

**(ii)** f(x) = x^{2} – 1, x = 1,−1

f(x) = x^{2} – 1

Given that x = (1 , -1)

Substitute x = 1 in f(x)

f(1) = 1^{2} – 1

= 1 – 1

= 0

Now, substitute x = (-1) in f(x)

f(-1) = (−1)^{2} – 1

= 1 – 1

= 0

*Since , the results when x = 1 and x = -1 are 0, so (1 , -1) are the roots of the polynomial f(x) = x ^{2 }– 1*

**(iii)** g(x) = 3x^{2} – 2 , x = 2/√3 , −2/√3

g(x) = 3x^{2} – 2

Substitute x = 2/√3 in g(x)

g(2/√3) = 3(2/√3)^{2} – 2

= 3(4/3) – 2

= 4 – 2

= 2 ≠ 0

Now, Substitute x = −2/√3 in g(x)

g(2/√3) = 3(-2/√3)^{2} – 2

= 3(4/3) – 2

= 4 – 2

= 2 ≠ 0

*The results when x = 2/√3 and x = −2/√3) are not 0. Therefore, (2/√3 , −2/√3 ) are not zeros of 3x ^{2}–2.*

**(iv)** p(x) = x^{3} – 6x^{2} + 11x – 6 , x = 1, 2, 3

p(1) = 1^{3} – 6(1)^{2} + 11x 1 – 6 = 1 – 6 + 11 – 6 = 0

p(2) = 2^{3} – 6(2)^{2} + 11×2 – 6 = 8 – 24 + 22 – 6 = 0

p(3) = 3^{3} – 6(3)^{2} + 11×3 – 6 = 27 – 54 + 33 – 6 = 0

*Therefore, x = 1, 2, 3 are zeros of p(x).*

**(v)** f(x) = 5x – π, x = 4/5

f(4/5) = 5 x 4/5 – π = 4 – π ≠ 0

*Therefore, x = 4/5 is not a zeros of f(x).*

**(vi)** f(x) = x^{2} , x = 0

f(0) = 0^{2} = 0

*Therefore, x = 0 is a zero of f(x).*

**(vii)** f(x) = lx + m, x = −m/l

f(−m/l) = l x −m/l + m = -m + m = 0

*Therefore, x = −m/l is a zero of f(x).*

**(viii)** f(x) = 2x + 1, x = ½

f(1/2) = 2x 1/2 + 1 = 1 + 1 = 2 ≠ 0

*Therefore, x = ½ is not a zero of f(x).*

### Exercise 6.3 Page No: 6.14

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

**Question 1: f(x) = x ^{3} + 4x^{2} – 3x + 10, g(x) = x + 4**

**Solution:**

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

Put g(x) =0

⇒ x + 4 = 0 or x = -4

Remainder = f(-4)

Now,

f(-4) = (-4)^{3} + 4(-4)^{2} – 3(-4) + 10 = -64 + 64 + 12 + 10 = *22*

**Actual Division: **

**Question 2: f(x) = 4x ^{4} – 3x^{3} – 2x^{2} + x – 7, g(x) = x – 1**

**Solution:**

f(x) = 4x^{4} – 3x^{3} – 2x^{2} + x – 7

Put g(x) =0

⇒ x – 1 = 0 or x = 1

Remainder = f(1)

Now,

f(1) = 4(1)^{4} – 3(1)^{3} – 2(1)^{2} + (1) – 7 = 4 – 3 – 2 + 1 – 7 = *-7*

**Actual Division: **

**Question 3: f(x) = 2x ^{4} – 6X^{3} + 2x^{2} – x + 2, g(x) = x + 2**

**Solution:**

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

Put g(x) = 0

⇒ x + 2 = 0 or x = -2

Remainder = f(-2)

Now,

f(-2) = 2(-2)^{4} – 6(-2)^{3} + 2(-2)^{2} – (-2) + 2 = 32 + 48 + 8 + 2 + 2 = *92*

**Actual Division: **

**Question 4: f(x) = 4x ^{3} – 12x^{2} + 14x – 3, g(x) = 2x – 1**

**Solution:**

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

Put g(x) =0

⇒ 2x -1 =0 or x = 1/2

Remainder = f(1/2)

Now,

f(1/2) = 4(1/2)^{3} – 12(1/2)^{2} + 14(1/2) – 3 = ½ – 3 + 7 – 3 = *3/2*

**Actual Division: **

**Question 5: f(x) = x ^{3} – 6x^{2} + 2x – 4, g(x) = 1 – 2x**

**Solution:**

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

Put g(x) = 0

⇒ 1 – 2x = 0 or x = 1/2

Remainder = f(1/2)

Now,

f(1/2) = (1/2)^{3} – 6(1/2)^{2} + 2(1/2) – 4 = 1 + 1/8 – 4 – 3/2 = *-35/8*

**Actual Division: **

**Question 6: f(x) = x ^{4} – 3x^{2} + 4, g(x) = x – 2**

**Solution:**

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

Put g(x) = 0

⇒ x – 2 = 0 or x = 2

Remainder = f(2)

Now,

f(2) = (2)^{4} – 3(2)^{2} + 4 = 16 – 12 + 4 = *8*

**Actual Division: **

**Question 7: f(x) = 9x ^{3} – 3x^{2} + x – 5, g(x) = x – 2/3**

**Solution:**

f(x) = 9x^{3} – 3x^{2} + x – 5, g(x) = x – 2/3

Put g(x) = 0

⇒ x – 2/3 = 0 or x = 2/3

Remainder = f(2/3)

Now,

f(2/3) = 9(2/3)^{3} – 3(2/3)^{2} + (2/3) – 5 = 8/3 – 4/3 + 2/3 – 5/1 = *-3*

**Actual Division: **

### Exercise 6.4 Page No: 6.24

**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)**

**Question 1: f(x) = x ^{3} – 6x^{2} + 11x – 6; g(x) = x – 3**

**Solution:**

If g(x) is a factor of f(x), then the remainder will be zero that is g(x) = 0.

g(x) = x -3 = 0

or x = 3

Remainder = f(3)

Now,

f(3) = (3)^{3} – 6(3)^{2} +11 x 3 – 6

= 27 – 54 + 33 – 6

= 60 – 60

= 0

*Therefore, g(x) is a factor of f(x)*

**Question 2: f(x) = 3X ^{4} + 17x^{3} + 9x^{2} – 7x – 10; g(x) = x + 5**

**Solution:**

If g(x) is a factor of f(x), then the remainder will be zero that is g(x) = 0.

g(x) = x + 5 = 0, then x = -5

Remainder = f(-5)

Now,

f(3) = 3(-5)^{4} + 17(-5)^{3} + 9(-5)^{2} – 7(-5) – 10

= 3 x 625 + 17 x (-125) + 9 x (25) – 7 x (-5) – 10

= 1875 -2125 + 225 + 35 – 10

= 0

*Therefore, g(x) is a factor of f(x).*

**Question 3: f(x) = x ^{5} + 3x^{4} – x^{3} – 3x^{2} + 5x + 15, g(x) = x + 3**

**Solution**:

If g(x) is a factor of f(x), then the remainder will be zero that is g(x) = 0.

g(x) = x + 3 = 0, then x = -3

Remainder = f(-3)

Now,

f(-3) = (-3)^{5} + 3(-3)^{4} – (-3)^{3} – 3(-3)^{2} + 5(-3) + 15

= -243 + 3 x 81 -(-27)-3 x 9 + 5(-3) + 15

= -243 +243 + 27-27- 15 + 15

= 0

*Therefore, g(x) is a factor of f(x).*

**Question 4: f(x) = x ^{3} – 6x^{2} – 19x + 84, g(x) = x – 7**

**Solution**:

If g(x) is a factor of f(x), then the remainder will be zero that is g(x) = 0.

g(x) = x – 7 = 0, then x = 7

Remainder = f(7)

Now,

f(7) = (7)^{3} – 6(7)^{2} – 19 x 7 + 84

= 343 – 294 – 133 + 84

= 343 + 84 – 294 – 133

= 0

*Therefore, g(x) is a factor of f(x).*

**Question 5: f(x) = 3x ^{3} + x^{2} – 20x + 12, g(x) = 3x – 2**

**Solution:**

If g(x) is a factor of f(x), then the remainder will be zero that is g(x) = 0.

g(x) = 3x – 2 = 0, then x = 2/3

Remainder = f(2/3)

Now,

f(2/3) = 3(2/3) ^{3} + (2/3) ^{2} – 20(2/3) + 12

= 3 x 8/27 + 4/9 – 40/3 + 12

= 8/9 + 4/9 – 40/3 + 12

= 0/9

= 0

*Therefore, g(x) is a factor of f(x).*

**Question 6: f(x) = 2x ^{3} – 9x^{2} + x + 12, g(x) = 3 – 2x**

**Solution:**

If g(x) is a factor of f(x), then the remainder will be zero that is g(x) = 0.

g(x) = 3 – 2x = 0, then x = 3/2

Remainder = f(3/2)

Now,

f(3/2) = 2(3/2)^{3} – 9(3/2)^{2} + (3/2) + 12

= 2 x 27/8 – 9 x 9/4 + 3/2 + 12

= 27/4 – 81/4 + 3/2 + 12

= 0/4

= 0

*Therefore, g(x) is a factor of f(x).*

**Question 7: f(x) = x ^{3} – 6x^{2} + 11x – 6, g(x) = x^{2} – 3x + 2**

**Solution:**

If g(x) is a factor of f(x), then the remainder will be zero that is g(x) = 0.

g(x) = 0

or x^{2} – 3x + 2 = 0

x^{2} – x – 2x + 2 = 0

x(x – 1) – 2(x – 1) = 0

(x – 1) (x – 2) = 0

Therefore x = 1 or x = 2

Now,

f(1) = (1)^{3} – 6(1)^{2} + 11(1) – 6 = 1-6+11-6= 12- 12 = 0

f(2) = (2)^{3} – 6(2)^{2} + 11(2) – 6 = 8 – 24 + 22 – 6 = 30 – 30 = 0

⇒ f(1) = 0 and f(2) = 0

*Which implies g(x) is factor of f(x).*

**Question 8: Show that (x – 2), (x + 3) and (x – 4) are factors of x ^{3} – 3x^{2} – 10x + 24.**

**Solution:**

Let f(x) = x^{3} – 3x^{2} – 10x + 24

If x – 2 = 0, then x = 2,

If x + 3 = 0 then x = -3,

and If x – 4 = 0 then x = 4

Now,

f(2) = (2)^{3} – 3(2)^{2} – 10 x 2 + 24 = 8 – 12 – 20 + 24 = 32 – 32 = 0

f(-3) = (-3)^{3} – 3(-3)^{2} – 10 (-3) + 24 = -27 -27 + 30 + 24 = -54 + 54 = 0

f(4) = (4)^{3} – 3(4)^{2} – 10 x 4 + 24 = 64-48 -40 + 24 = 88 – 88 = 0

f(2) = 0

f(-3) = 0

f(4) = 0

*Hence (x – 2), (x + 3) and (x – 4) are the factors of f(x)*

**Question 9: Show that (x + 4), (x – 3) and (x – 7) are factors of x ^{3} – 6x^{2} – 19x + 84.**

**Solution:**

Let f(x) = x^{3} – 6x^{2} – 19x + 84

If x + 4 = 0, then x = -4

If x – 3 = 0, then x = 3

and if x – 7 = 0, then x = 7

Now,

f(-4) = (-4)^{3} – 6(-4)^{2} – 19(-4) + 84 = -64 – 96 + 76 + 84 = 160 – 160 = 0

f(-4) = 0

f(3) = (3)^{ 3} – 6(3)^{ 2} – 19 x 3 + 84 = 27 – 54 – 57 + 84 = 111 -111=0

f(3) = 0

f(7) = (7)^{ 3} – 6(7)^{ 2} – 19 x 7 + 84 = 343 – 294 – 133 + 84 = 427 – 427 = 0

f(7) = 0

*Hence (x + 4), (x – 3), (x – 7) are the factors of f(x).*

### Exercise 6.5 Page No: 6.32

**Using factor theorem, factorize each of the following polynomials:**

**Question 1: x ^{3} + 6x^{2} + 11x + 6**

**Solution**:

Let f(x) = x^{3} + 6x^{2} + 11x + 6

Step 1: Find the factors of constant term

Here constant term = 6

Factors of 6 are ±1, ±2, ±3, ±6

Step 2: Find the factors of f(x)

Let x + 1 = 0

⇒ x = -1

Put the value of x in f(x)

f(-1) = (−1)^{3} + 6(−1)^{2} + 11(−1) + 6

= -1 + 6 -11 + 6

= 12 – 12

= 0

So, (x + 1) is the factor of f(x)

Let x + 2 = 0

⇒ x = -2

Put the value of x in f(x)

f(-2) = (−2)^{3} + 6(−2)^{2} + 11(−2) + 6 = -8 + 24 – 22 + 6 = 0

So, (x + 2) is the factor of f(x)

Let x + 3 = 0

⇒ x = -3

Put the value of x in f(x)

f(-3) = (−3)^{3} + 6(−3)^{2} + 11(−3) + 6 = -27 + 54 – 33 + 6 = 0

So, (x + 3) is the factor of f(x)

*Hence, f(x) = (x + 1)(x + 2)(x + 3)*

**Question 2: x ^{3} + 2x^{2} – x – 2**

**Solution**:

Let f(x) = x^{3} + 2x^{2} – x – 2

Constant term = -2

Factors of -2 are ±1, ±2

Let x – 1 = 0

⇒ x = 1

Put the value of x in f(x)

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

So, (x – 1) is factor of f(x)

Let x + 1 = 0

⇒ x = -1

Put the value of x in f(x)

f(-1) = (-1)^{3} + 2(-1)^{2} – 1 – 2 = -1 + 2 + 1 – 2 = 0

(x + 1) is a factor of f(x)

Let x + 2 = 0

⇒ x = -2

Put the value of x in f(x)

f(-2) = (-2)^{3} + 2(-2)^{2} – (-2) – 2 = -8 + 8 + 2 – 2 = 0

(x + 2) is a factor of f(x)

Let x – 2 = 0

⇒ x = 2

Put the value of x in f(x)

f(2) = (2)^{3} + 2(2)^{2} – 2 – 2 = 8 + 8 – 2 – 2 = 12 ≠ 0

(x – 2) is not a factor of f(x)

*Hence f(x) = (x + 1)(x- 1)(x+2)*

**Question 3: x ^{3} – 6x^{2} + 3x + 10**

**Solution**:

Let f(x) = x^{3} – 6x^{2} + 3x + 10

Constant term = 10

Factors of 10 are ±1, ±2, ±5, ±10

Let x + 1 = 0 or x = -1

f(-1) = (-1)^{3} – 6(-1)^{2} + 3(-1) + 10 = 10 – 10 = 0

f(-1) = 0

Let x + 2 = 0 or x = -2

f(-2) = (-2)^{3} – 6(-2)^{2} + 3(-2) + 10 = -8 – 24 – 6 + 10 = -28

f(-2) ≠ 0

Let x – 2 = 0 or x = 2

f(2) = (2)^{3} – 6(2)^{2} + 3(2) + 10 = 8 – 24 + 6 + 10 = 0

f(2) = 0

Let x – 5 = 0 or x = 5

f(5) = (5)^{3} – 6(5)^{2} + 3(5) + 10 = 125 – 150 + 15 + 10 = 0

f(5) = 0

Therefore, (x + 1), (x – 2) and (x-5) are factors of f(x)

*Hence f(x) = (x + 1) (x – 2) (x-5)*

**Question 4: x ^{4} – 7x^{3} + 9x^{2} + 7x- 10**

**Solution:**

Let f(x) = x^{4} – 7x^{3} + 9x^{2} + 7x- 10

Constant term = -10

Factors of -10 are ±1, ±2, ±5, ±10

Let x – 1 = 0 or x = 1

f(1) = (1)^{4} – 7(1)^{3} + 9(1)^{2} + 7(1) – 10 = 1 – 7 + 9 + 7 -10 = 0

f(1) = 0

Let x + 1 = 0 or x = -1

f(-1) = (-1)^{4} – 7(-1)^{3} + 9(-1)^{2} + 7(-1) – 10 = 1 + 7 + 9 – 7 -10 = 0

f(-1) = 0

Let x – 2 = 0 or x = 2

f(2) = (2)^{4} – 7(2)^{3} + 9(2)^{2} + 7(2) – 10 = 16 – 56 + 36 + 14 – 10 = 0

f(2) = 0

Let x – 5 = 0 or x = 5

f(5) = (5)^{4} – 7(5)^{3} + 9(5)^{2} + 7(5) – 10 = 625 – 875 + 225 + 35 – 10 = 0

f(5) = 0

Therefore, (x – 1), (x + 1), (x – 2) and (x-5) are factors of f(x)

*Hence f(x) = (x – 1) (x + 1) (x – 2) (x-5)*

**Question 5: x ^{4} – 2x^{3} – 7x^{2} + 8x + 12**

**Solution**:

f(x) = x^{4} – 2x^{3} – 7x^{2} + 8x + 12

Constant term = 12

Factors of 12 are ±1, ±2, ±3, ±4, ±6, ±12

Let x – 1 = 0 or x = 1

f(1) = (1)^{4} – 2(1)^{3} – 7(1)^{2} + 8(1) + 12 = 1 – 2 – 7 + 8 + 12 = 12

f(1) ≠ 0

Let x + 1 = 0 or x = -1

f(-1) = (-1)^{4} – 2(-1)^{3} – 7(-1)^{2} + 8(-1) + 12 = 1 + 2 – 7 – 8 + 12 = 0

f(-1) = 0

Let x +2 = 0 or x = -2

f(-2) = (-2)^{4} – 2(-2)^{3} – 7(-2)^{2} + 8(-2) + 12 = 16 + 16 – 28 – 16 + 12 = 0

f(-2) = 0

Let x – 2 = 0 or x = 2

f(2) = (2)^{4} – 2(2)^{3} – 7(2)^{2} + 8(2) + 12 = 16 – 16 – 28 + 16 + 12 = 0

f(2) = 0

Let x – 3 = 0 or x = 3

f(3) = (3)^{4} – 2(3)^{3} – 7(3)^{2} + 8(3) + 12 = 0

f(3) = 0

Therefore, (x + 1), (x + 2), (x – 2) and (x-3) are factors of f(x)

*Hence f(x) = (x + 1)(x + 2) (x – 2) (x-3)*

**Question 6: x ^{4} + 10x^{3} + 35x^{2} + 50x + 24**

**Solution**:

Let f(x) = x^{4} + 10x^{3} + 35x^{2} + 50x + 24

Constant term = 24

Factors of 24 are ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24

Let x + 1 = 0 or x = -1

f(-1) = (-1)^{4} + 10(-1)^{3} + 35(-1)^{2} + 50(-1) + 24 = 1 – 10 + 35 – 50 + 24 = 0

f(1) = 0

(x + 1) is a factor of f(x)

Likewise, (x + 2),(x + 3),(x + 4) are also the factors of f(x)

*Hence f(x) = (x + 1) (x + 2)(x + 3)(x + 4)*

**Question 7: 2x ^{4} – 7x^{3} – 13x^{2} + 63x – 45**

**Solution:**

Let f(x) = 2x^{4} – 7x^{3} – 13x^{2} + 63x – 45

Constant term = -45

Factors of -45 are ±1, ±3, ±5, ±9, ±15, ±45

Here coefficient of x^4 is 2. So possible rational roots of f(x) are

±1, ±3, ±5, ±9, ±15, ±45, ±1/2,±3/2,±5/2,±9/2,±15/2,±45/2

Let x – 1 = 0 or x = 1

f(1) = 2(1)^{4} – 7(1)^{3} – 13(1)^{2} + 63(1) – 45 = 2 – 7 – 13 + 63 – 45 = 0

f(1) = 0

f(x) can be written as,

f(x) = (x-1) (2x^{3} – 5x^{2} -18x +45)

or f(x) =(x-1)g(x) …(1)

Let x – 3 = 0 or x = 3

f(3) = 2(3)^{4} – 7(3)^{3} – 13(3)^{2} + 63(3) – 45 = = 162 – 189 – 117 + 189 – 45= 0

f(3) = 0

Now, we are available with 2 factors of f(x), (x – 1) and (x – 3)

Here g(x) = 2x^{2} (x-3) + x(x-3) -15(x-3)

Taking (x-3) as common

= (x-3)(2x^{2} + x – 15)

= (x-3)(2x^{2}+6x – 5x -15)

= (x-3)(2x-5)(x+3)

= (x-3)(x+3)(2x-5) ….(2)

From (1) and (2)

*f(x) =(x-1) (x-3)(x+3)(2x-5)*

### Exercise VSAQs Page No: 6.33

**Question 1: Define zero or root of a polynomial**

**Solution**:

zero or root, is a solution to the polynomial equation, f(y) = 0.

It is that value of y that makes the polynomial equal to *zero.*

**Question 2: If x = 1/2 is a zero of the polynomial f(x) = 8x ^{3} + ax^{2} – 4x + 2, find the value of a.**

**Solution: **

If x = 1/2 is a zero of the polynomial f(x), then f(1/2) = 0

8(1/2)^{3} + a(1/2)^{2} – 4(1/2) + 2 = 0

8 x 1/8 + a/4 – 2 + 2 = 0

1 + a/4 = 0

*a = -4*

**Question 3: Write the remainder when the polynomial f(x) = x ^{3} + x^{2} – 3x + 2 is divided by x + 1.**

**Solution: **

Using factor theorem,

Put x + 1 = 0 or x = -1

f(-1) is the remainder.

Now,

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

= -1 + 1 + 3 + 2

= 5

*Therefore 5 is the remainder.*

**Question 4: Find the remainder when x ^{3} + 4x^{2} + 4x-3 if divided by x**

**Solution: **

Using factor theorem,

Put x = 0

f(0) is the remainder.

Now,

f(0) = 0^{3} + 4(0)^{2} + 4×0 -3 = -3

*Therefore -3 is the remainder.*

**Question 5: **If x+1 is a factor of x^{3} + a, then write the value of a.

**Solution: **

Let f(x) = x^{3} + a

If x+1 is a factor of x^{3} + a then f(-1) = 0

(-1)^{3} + a = 0

-1 + a = 0

*or a = 1*

**Question 6: 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.**

Solution:

From the statement, we have f(1) = 6

(1)^4 – 2(1)^{3} + 3(1)^{2} – a(1) – b = 6

1 – 2 + 3 – a – b = 6

2 – a – b = 6

*a + b = -4*

**RD Sharma Solutions for Class 9 Maths Chapter 6 Factorization of Polynomials**

In the 6th Chapter of **Class 9 RD Sharma Solutions,** students will study important concepts on the Factorization of Polynomials, as listed below.

- Factorization of Polynomials Introduction
- Terms and Coefficients
- Degree of a Polynomial
- Types of Polynomials

- Monomial: A Polynomial containing only one term.
- Binomial: A Polynomial containing two terms.
- Trinomial: A Polynomial containing three terms.

- Remainder Theorem
- Factorization of Polynomials by Using the Factor Theorem

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