# ICSE Solutions for Selina Concise Chapter 8 Remainder and Factor Theorem Class 10 Maths

### Exercise 8(A)

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

∴ (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

∴ (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

∴ (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 = 3/2 = 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

∴ n = 7

### Exercise 8(B)

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

∴ (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

∴ (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

∴ (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) 3x+ 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) = 3x+ 2x2 − 19x + 6

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

= 3(2)+ 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) (3x– 4x – 15)

= (x + 2) (3x– 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

∴  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

∴ (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)

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

∴ 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).

∴ 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).

∴ 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