Selina Solutions Concise Maths Class 10 Chapter 8 Remainder and Factor Theorems

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Exercises of Concise Selina Solutions Class 10 Maths Chapter 8 Remainder and Factor Theorems

Exercise 8(A) Solutions

Exercise 8(B) Solutions

Exercise 8(C) Solutions

Access Selina Solutions Concise Maths Class 10 Chapter 8 Remainder and Factor Theorems

Exercise 8(A) Page No: 108

1. Find, in each case, the remainder when:

(i) x⁴ – 3x² + 2x + 1 is divided by x – 1.

(ii) x³ + 3x² – 12x + 4 is divided by x – 2.

(ii) x⁴ + 1 is divided by x + 1.

Solution:

From remainder theorem, we know that when a polynomial f (x) is divided by (x – a), then the remainder is f(a).

(i) Given, f(x) = x⁴ – 3x² + 2x + 1 is divided by x – 1

So, remainder = f(1) = (1)⁴ – 3(1)² + 2(1) + 1 = 1 – 3 + 2 + 1 = 1

(ii) Given, f(x) = x³ + 3x² – 12x + 4 is divided by x – 2

So, remainder = f(2) = (2)³ + 3(2)² – 12(2) + 4 = 8 + 12 – 24 + 4 = 0

(iii) Given, f(x) = x⁴ + 1 is divided by x + 1

So, remainder = f(-1) = (-1)⁴ + 1 = 2

2. Show that:

(i) x – 2 is a factor of 5x² + 15x – 50

(ii) 3x + 2 is a factor of 3x² – x – 2

Solution:

(x – a) is a factor of a polynomial f(x) if the remainder, when f(x) is divided by (x – a), is 0, i.e., if f(a) = 0.

(i) f(x) = 5x² + 15x – 50

f(2) = 5(2)² + 15(2) – 50 = 20 + 30 – 50 = 0

As the remainder is zero for x = 2

Thus, we can conclude that (x – 2) is a factor of 5x² + 15x – 50

(ii) f(x) = 3x² – x – 2

f(-2/3) = 3(-2/3)² – (-2/3) – 2 = 4/3 + 2/3 – 2 = 2 – 2 = 0

As the remainder is zero for x = -2/3

Thus, we can conclude that (3x + 2) is a factor of 3x² – x – 2

3. Use the Remainder Theorem to find which of the following is a factor of 2x³ + 3x² – 5x – 6.

(i) x + 1 (ii) 2x – 1

(iii) x + 2

Solution:

From remainder theorem we know that when a polynomial f (x) is divided by x – a, then the remainder is f(a).

Here, f(x) = 2x³ + 3x² – 5x – 6

(i) f (-1) = 2(-1)³ + 3(-1)² – 5(-1) – 6 = -2 + 3 + 5 – 6 = 0

⇒ Remainder is zero for x = -1

Therefore, (x + 1) is a factor of the polynomial f(x).

(ii) f(1/2) = 2(1/2)³ + 3(1/2)² – 5(1/2) – 6

= ¼ + ¾ – 5/2 – 6

= -5/2 – 5 = -15/2

⇒ Remainder is not equals to zero for x = 1/2

Therefore, (2x – 1) is not a factor of the polynomial f(x).

(iii) f (-2) = 2(-2)³ + 3(-2)² – 5(-2) – 6 = -16 + 12 + 10 – 6 = 0

⇒ Remainder is zero for x = -2

Therefore, (x + 2) is a factor of the polynomial f(x).

4. (i) If 2x + 1 is a factor of 2x² + ax – 3, find the value of a.

(ii) Find the value of k, if 3x – 4 is a factor of expression 3x² + 2x – k.

Solution:

(i) Given, 2x + 1 is a factor of f(x) = 2x² + ax – 3.

So, f(-1/2) = 0

2(-1/2)² + a(-1/2) – 3 = 0

½ – a/2 – 3 = 0

1 – a – 6 = 0

a = -5

(ii) Given, 3x – 4 is a factor of g(x) = 3x² + 2x – k.

So, f(4/3) = 0

3(4/3)² + 2(4/3) – k = 0

16/3 + 8/3 – k = 0

24/3 = k

k = 8

5. Find the values of constants a and b when x – 2 and x + 3 both are the factors of expression x³ + ax² + bx – 12.

Solution:

Here, f(x) = x³ + ax² + bx – 12

Given, x – 2 and x + 3 both are the factors of f(x)

So,

f(2) and f(-3) both should be equal to zero.

f(2) = (2)³ + a(2)² + b(2) – 12

0 = 8 + 4a + 2b – 12

0 = 4a + 2b – 4

2a + b = 2 …. (1)

Now,

f(-3) = (-3)³ + a(-3)² + b(-3) – 12

0 = -27 + 9a – 3b – 12

9a – 3b – 39 = 0

3a – b = 13 …. (2)

Adding (1) and (2), we get,

5a = 15

Thus, a = 3

Putting the value of a in (1), we have

6 + b = 2

Thus, b = -4

6. Find the value of k, if 2x + 1 is a factor of (3k + 2)x³ + (k – 1). 

Solution:

Let take f(x) = (3k + 2)x³ + (k – 1)

Now, 2x + 1 = 0

x = -1/2

As, 2x + 1 is a factor of f(x) then the remainder should be 0.

f(-1/2) = (3k + 2)(-1/2)³ + (k – 1) = 0

5k = 10 = 0

k = 2

7. Find the value of a, if x – 2 is a factor of 2x⁵ – 6x⁴ – 2ax³ + 6ax² + 4ax + 8.

Solution:

Given, f(x) = 2x⁵ – 6x⁴ – 2ax³ + 6ax² + 4ax + 8 and x – 2 is a factor of f(x).

So, x – 2 = 0; x = 2

Hence, f(2) = 0

2(2)⁵ – 6(2)⁴ – 2a(2)³ + 6a(2)² + 4a(2) + 8 = 0

64 – 96 – 16a + 24a + 8a + 8 = 0

-24 + 16a = 0

16a = 24

Thus, a = 1.5

8. Find the values of m and n so that x – 1 and x + 2 both are factors of x³ + (3m + 1) x² + nx – 18.

Solution:

Let f(x) = x³ + (3m + 1) x² + nx – 18

Given, (x – 1) and (x + 2) are the factors of f(x).

So,

x – 1 = 0; x = 1 and x + 2 = 0; x = -2

f(1) and f(-2) both should be equal to zero.

(1)³ + (3m + 1)(1)² + n(1) – 18 = 0

1 + 3m + 1 + n – 18 = 0

3m + n – 16 = 0 ….. (1)

And,

(-2)³ + (3m + 1)(-2)² + n(-2) – 18 = 0

8 + 12m + 4 – 2n – 18 = 0

12m – 2n – 22 = 0

6m – n – 11 = 0 ….. (2)

Adding (1) and (2), we get,

9m – 27 = 0

Thus, m = 3

Putting the value of m in (1), we have

3(3) + n – 16 =0

9 + n – 16 = 0

Therefore, n = 7

Exercise 8(B) Page No: 111

1. Using the Factor Theorem, show that:

(i) (x – 2) is a factor of x³ – 2x² – 9x + 18. Hence, factorise the expression x³ – 2x² – 9x + 18 completely.

(ii) (x + 5) is a factor of 2x³ + 5x² – 28x – 15. Hence, factorise the expression 2x³ + 5x² – 28x – 15 completely.

(iii) (3x + 2) is a factor of 3x³ + 2x² – 3x – 2. Hence, factorise the expression 3x³ + 2x² – 3x – 2 completely.

Solution:

(i) Here, f(x) = x³ – 2x² – 9x + 18

So, x – 2 = 0 ⇒ x = 2

Thus, remainder = f(2)

= (2)³ – 2(2)² – 9(2) + 18

= 8 – 8 – 18 + 18

= 0

Therefore, (x – 2) is a factor of f(x).

Now, performing division of polynomial f(x) by (x – 2) we have

Thus, x³ – 2x² – 9x + 18 = (x – 2) (x² – 9) = (x – 2) (x + 3) (x – 3)

(ii) Here, f(x) = 2x³ + 5x² – 28x – 15

So, x + 5 = 0 ⇒ x = -5

Thus, remainder = f(-5)

= 2(-5)³ + 5(-5)² – 28(-5) – 15

= -250 + 125 + 140 – 15

= -265 + 265

= 0

Therefore, (x + 5) is a factor of f(x).

Now, performing division of polynomial f(x) by (x + 5) we get

So, 2x³ + 5x² – 28x – 15 = (x + 5) (2x² – 5x – 3)

Further, on factorisation

= (x + 5) [2x² – 6x + x – 3]

= (x + 5) [2x(x – 3) + 1(x – 3)] = (x + 5) (2x + 1) (x – 3)

Thus, f(x) is factorised as (x + 5) (2x + 1) (x – 3)

(iii) Here, f(x) = 3x³ + 2x² – 3x – 2

So, 3x + 2 = 0 ⇒ x = -2/3

Thus, remainder = f(-2/3)

= 3(-2/3)³ + 2(-2/3)² – 3(-2/3) – 2

= -8/9 + 8/9 + 2 – 2

= 0

Therefore, (3x + 2) is a factor of f(x).

Now, performing division of polynomial f(x) by (3x + 2) we get

Thus, 3x³ + 2x² – 3x – 2 = (3x + 2) (x² – 1) = (3x + 2) (x – 1) (x + 1)

2. Using the Remainder Theorem, factorise each of the following completely.

(i) 3x³ + 2x² − 19x + 6

(ii) 2x³ + x² – 13x + 6

(iii) 3x³ + 2x² – 23x – 30

(iv) 4x³ + 7x² – 36x – 63

(v) x³ + x² – 4x – 4

Solution:

(i) Let f(x) = 3x³ + 2x² − 19x + 6

For x = 2, the value of f(x) will be

= 3(2)³ + 2(2)² – 19(2) + 6

= 24 + 8 – 38 + 6 = 0

As f(2) = 0, so (x – 2) is a factor of f(x).

Now, performing long division we have

Thus, f(x) = (x -2) (3x² + 8x – 3)

= (x – 2) (3x² + 9x – x – 3)

= (x – 2) [3x(x + 3) -1(x + 3)]

= (x – 2) (x + 3) (3x – 1)

(ii) Let f(x) = 2x³ + x² – 13x + 6

For x = 2, the value of f(x) will be

f(2) = 2(2)³ + (2)² – 13(2) + 6 = 16 + 4 – 26 + 6 = 0

As f(2) = 0, so (x – 2) is a factor of f(x).

Now, performing long division we have

Thus, f(x) = (x -2) (2x² + 5x – 3)

= (x – 2) [2x² + 6x – x – 3]

= (x – 2) [2x(x + 3) -1(x + 3)]

= (x – 2) [2x(x + 3) -1(x + 3)]

= (x – 2) (2x – 1) (x + 3)

(iii) Let f(x) = 3x³ + 2x² – 23x – 30

For x = -2, the value of f(x) will be

f(-2) = 3(-2)³ + 2(-2)² – 23(-2) – 30

= -24 + 8 + 46 – 30 = -54 + 54 = 0

As f(-2) = 0, so (x + 2) is a factor of f(x).

Now, performing long division we have

Thus, f(x) = (x + 2) (3x² – 4x – 15)

= (x + 2) (3x² – 9x + 5x – 15)

= (x + 2) [3x(x – 3) + 5(x – 3)]

= (x + 2) (3x + 5) (x – 3)

(iv) Let f(x) = 4x³ + 7x² – 36x – 63

For x = 3, the value of f(x) will be

f(3) = 4(3)³ + 7(3)² – 36(3) – 63

= 108 + 63 – 108 – 63 = 0

As f(3) = 0, (x + 3) is a factor of f(x).

Now, performing long division we have

Thus, f(x) = (x + 3) (4x² – 5x – 21)

= (x + 3) (4x² – 12x + 7x – 21)

= (x + 3) [4x(x – 3) + 7(x – 3)]

= (x + 3) (4x + 7) (x – 3)

(v) Let f(x) = x³ + x² – 4x – 4

For x = -1, the value of f(x) will be

f(-1) = (-1)³ + (-1)² – 4(-1) – 4

= -1 + 1 + 4 – 4 = 0

As, f(-1) = 0 so (x + 1) is a factor of f(x).

Now, performing long division we have

Thus, f(x) = (x + 1) (x² – 4)

= (x + 1) (x – 2) (x + 2)

3. Using the Remainder Theorem, factorise the expression 3x³ + 10x² + x – 6. Hence, solve the equation 3x³ + 10x² + x – 6 = 0.

Solution:

Let’s take f(x) = 3x³ + 10x² + x – 6

For x = -1, the value of f(x) will be

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

As, f(-1) = 0 so (x + 1) is a factor of f(x).

Now, performing long division we have

Thus, f(x) = (x + 1) (3x² + 7x – 6)

= (x + 1) (3x² + 9x – 2x – 6)

= (x + 1) [3x(x + 3) -2(x + 3)]

= (x + 1) (x + 3) (3x – 2)

Now, 3x³ + 10x² + x – 6 = 0

(x + 1) (x + 3) (3x – 2) = 0

Therefore,

x = -1, -3 or 2/3

4. Factorise the expression f (x) = 2x³ – 7x² – 3x + 18. Hence, find all possible values of x for which f(x) = 0.

Solution:

Let f(x) = 2x³ – 7x² – 3x + 18

For x = 2, the value of f(x) will be

f(2) = 2(2)³ – 7(2)² – 3(2) + 18

= 16 – 28 – 6 + 18 = 0

As f(2) = 0, (x – 2) is a factor of f(x).

Now, performing long division we have

Thus, f(x) = (x – 2) (2x² – 3x – 9)

= (x – 2) (2x² – 6x + 3x – 9)

= (x – 2) [2x(x – 3) + 3(x – 3)]

= (x – 2) (x – 3) (2x + 3)

Now, for f(x) = 0

(x – 2) (x – 3) (2x + 3) = 0

Hence x = 2, 3 or -3/2

5. Given that x – 2 and x + 1 are factors of f(x) = x³ + 3x² + ax + b; calculate the values of a and b. Hence, find all the factors of f(x).

Solution:

Let f(x) = x³ + 3x² + ax + b

As, (x – 2) is a factor of f(x), so f(2) = 0

(2)³ + 3(2)² + a(2) + b = 0

8 + 12 + 2a + b = 0

2a + b + 20 = 0 … (1)

And as, (x + 1) is a factor of f(x), so f(-1) = 0

(-1)³ + 3(-1)² + a(-1) + b = 0

-1 + 3 – a + b = 0

-a + b + 2 = 0 … (2)

Subtracting (2) from (1), we have

3a + 18 = 0

a = -6

On substituting the value of a in (ii), we have

b = a – 2 = -6 – 2 = -8

Thus, f(x) = x³ + 3x² – 6x – 8

Now, for x = -1

f(-1) = (-1)³ + 3(-1)² – 6(-1) – 8 = -1 + 3 + 6 – 8 = 0

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

Now, performing long division we have

Hence, f(x) = (x + 1) (x² + 2x – 8)

= (x + 1) (x² + 4x – 2x – 8)

= (x + 1) [x(x + 4) – 2(x + 4)]

= (x + 1) (x + 4) (x – 2)

Exercise 8(C) Page No: 112

1. Show that (x – 1) is a factor of x³ – 7x² + 14x – 8. Hence, completely factorise the given expression.

Solution:

Let f(x) = x³ – 7x² + 14x – 8

Then, for x = 1

f(1) = (1)³ – 7(1)² + 14(1) – 8 = 1 – 7 + 14 – 8 = 0

Thus, (x – 1) is a factor of f(x).

Now, performing long division we have

Hence, f(x) = (x – 1) (x² – 6x + 8)

= (x – 1) (x² – 4x – 2x + 8)

= (x – 1) [x(x – 4) -2(x – 4)]

= (x – 1) (x – 4) (x – 2)

2. Using Remainder Theorem, factorise:

 x³ + 10x² – 37x + 26 completely.

Solution:

Let f(x) = x³ + 10x² – 37x + 26

From remainder theorem, we know that

For x = 1, the value of f(x) is the remainder

f(1) = (1)³ + 10(1)² – 37(1) + 26 = 1 + 10 – 37 + 26 = 0

As f(1) = 0, x – 1 is a factor of f(x).

Now, performing long division we have

Thus, f(x) = (x – 1) (x² + 11x – 26)

= (x – 1) (x² + 13x – 2x – 26)

= (x – 1) [x(x + 13) – 2(x + 13)]

= (x – 1) (x + 13) (x – 2)

3. When x³ + 3x² – mx + 4 is divided by x – 2, the remainder is m + 3. Find the value of m.

Solution:

Let f(x) = x³ + 3x² – mx + 4

From the question, we have

f(2) = m + 3

(2)³ + 3(2)² – m(2) + 4 = m + 3

8 + 12 – 2m + 4 = m + 3

24 – 3 = m + 2m

3m = 21

Thus, m = 7

4. What should be subtracted from 3x³ – 8x² + 4x – 3, so that the resulting expression has x + 2 as a factor?

Solution:

Let’s assume the required number to be k.

And let f(x) = 3x³ – 8x² + 4x – 3 – k

From the question, we have

f(-2) = 0

3(-2)³ – 8(-2)² + 4(-2) – 3 – k = 0

-24 – 32 – 8 – 3 – k = 0

-67 – k = 0

k = -67

Therefore, the required number is -67.

5. If (x + 1) and (x – 2) are factors of x³ + (a + 1)x² – (b – 2)x – 6, find the values of a and b. And then, factorise the given expression completely.

Solution:

Let’s take f(x) = x³ + (a + 1)x² – (b – 2)x – 6

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

Then, remainder = f(-1) = 0

(-1)³ + (a + 1)(-1)² – (b – 2) (-1) – 6 = 0

-1 + (a + 1) + (b – 2) – 6 = 0

a + b – 8 = 0 … (1)

And as, (x – 2) is a factor of f(x).

Then, remainder = f(2) = 0

(2)³ + (a + 1) (2)² – (b – 2) (2) – 6 = 0

8 + 4a + 4 – 2b + 4 – 6 = 0

4a – 2b + 10 = 0

2a – b + 5 = 0 … (2)

Adding (1) and (2), we get

3a – 3 = 0

Thus, a = 1

Substituting the value of a in (i), we get,

1 + b – 8 = 0

Thus, b = 7

Hence, f(x) = x³ + 2x² – 5x – 6

Now as (x + 1) and (x – 2) are factors of f(x).

So, (x + 1) (x – 2) = x² – x – 2 is also a factor of f(x).

Therefore, f(x) = x³ + 2x² – 5x – 6 = (x + 1) (x – 2) (x + 3)

6. If x – 2 is a factor of x² + ax + b and a + b = 1, find the values of a and b.

Solution:

Let f(x) = x² + ax + b

Given, (x – 2) is a factor of f(x).

Then, remainder = f(2) = 0

(2)² + a(2) + b = 0

4 + 2a + b = 0

2a + b = -4 … (1)

And also given that,

a + b = 1 … (2)

Subtracting (2) from (1), we have

a = -5

On substituting the value of a in (2), we have

b = 1 – (-5) = 6

7. Factorise x³ + 6x² + 11x + 6 completely using factor theorem.

Solution:

Let f(x) = x³ + 6x² + 11x + 6

For x = -1, the value of f(x) is

f(-1) = (-1)³ + 6(-1)² + 11(-1) + 6

= -1 + 6 – 11 + 6 = 12 – 12 = 0

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

Therefore, f(x) = (x + 1) (x² + 5x + 6)

= (x + 1) (x² + 3x + 2x + 6)

= (x + 1) [x(x + 3) + 2(x + 3)]

= (x + 1) (x + 3) (x + 2)

8. Find the value of ‘m’, if mx³ + 2x² – 3 and x² – mx + 4 leave the same remainder when each is divided by x – 2.

Solution:

Let f(x) = mx³ + 2x² – 3 and g(x) = x² – mx + 4

From the question, it’s given that f(x) and g(x) leave the same remainder when divided by (x – 2). So, we have:

f(2) = g(2)

m(2)³ + 2(2)² – 3 = (2)² – m(2) + 4

8m + 8 – 3 = 4 – 2m + 4

10m = 3

Thus, m = 3/10

The given solutions are as per the 2019-20 Concise Selina textbook. The Selina Solutions for the academic year 2023-24 will be updated soon.