# Class 10 SELINA Solutions Maths Chapter 8 - Remainder And Factor Theorems

## Remainder And Factor Theorems Exercise Ex. 8(A)

### Solution 1(a)

Correct option: (i) –1

x – 1 = 0 ⇒ x = 1

Since x – 1 is a factor of 8x^{2} – 7x + m,

Remainder = 0

⇒ 8(1)^{2} – 7(1) + m = 0

⇒ 8 – 7 + m = 0

⇒ m = –1

### Solution 1(b)

Correct option: (ii) (x – 2)

(x – 2)^{2} – (x^{2} – 4)

= (x – 2)^{2} – (x – 2)(x + 2)

= (x – 2)(x – 2 – x – 2)

= (x – 2)( –4)

Hence, (x – 2) is a factor of a given polynomial.

### Solution 1(c)

Correct option: (iii) 6

x – 1 = 0 ⇒ x = 1

Since x – 1 is a factor of x^{3} – kx^{2} + 11x – 6,

Remainder = 0

⇒ (1)^{3} – k(1)^{2} + 11(1) – 6 = 0

⇒ 1 – k + 11 – 6 = 0

⇒ 6 – k = 0

⇒ k = 6

### Solution 1(d)

Correct option: (iv) -5

x – a = 0 ⇒ x = a

Since x – a is a factor of x^{3} – ax^{2} + x + 5,

Remainder = 0

⇒ (a)^{3} – a(a)^{2} + a + 5 = 0

⇒ a^{3} – a^{3} + a + 5 = 0

⇒ a = –5

### Solution 1(e)

Correct option: (i) x^{3} – x^{2} + x – 6

x – 2 = 0 ⇒ x = 2

Consider polynomial x^{3} – x^{2} + x – 6.

When x^{3} – x^{2} + x – 6 is divided by (x – 2),

Remainder = (2)^{3} – (2)^{2} + 2 – 6

= 8 – 4 + 2 – 6

= 0

Hence, (x – 2) is a factor of x^{3} – x^{2} + x – 6.

### Solution 2

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

### Solution 3

(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.

### Solution 4

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

Let f(x) = 2x^{3} + 3x^{2} - 5x - 6

(i) f (-1) = 2(-1)^{3} + 3(-1)^{2} - 5(-1) - 6 = -2 + 3 + 5 - 6 = 0

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

(ii)

Thus, (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

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

### Solution 5

(i) 2x + 1 is a factor of f(x) = 2x^{2} + ax - 3.

(ii) 3x - 4 is a factor of g(x) = 3x^{2} + 2x - k.

### Solution 6

Let f(x) = x^{3} + ax^{2} + bx - 12

x - 2 = 0 x = 2

x - 2 is a factor of f(x). So, remainder = 0

x + 3 = 0 x = -3

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

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

5a - 15 = 0

a = 3

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

6 + b - 2 = 0

b = -4

### Solution 7

Let f(x) = (3k + 2)x^{3} + (k - 1)

2x + 1 = 0

Since, 2x + 1 is a factor of f(x), remainder is 0.

### Solution 8

Let f(x) = x^{3} + (3m +
1) x^{2} + nx - 18

x - 1 = 0 x = 1

x - 1 is a factor of f(x). So, remainder = 0

x + 2 = 0 x = -2

x + 2 is a factor of f(x). So, remainder = 0

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

9m - 27 = 0

m = 3

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

3(3) + n - 16 =0

9 + n - 16 = 0

n = 7

### Solution 9

Let f(x) = x^{3} + 2x^{2}
- kx + 4

x - 2 = 0 x = 2

On dividing f(x) by x - 2, it leaves a remainder k.

### Solution 10

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

x + 3 = 0 x = -3

On dividing f(x) by x + 3, it leaves a remainder 5.

### Solution 11

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

x - 2 = 0 x = 2

Since, x - 2 is a factor, remainder = 0

x - 3 = 0 x = 3

On dividing f(x) by x - 3, it leaves a remainder 3.

Subtracting (i) from (ii), we get,

a + 3 = 0

a = -3

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

-6 + b + 7 = 0

b = -1

### Solution 12

Let the number k be added and the resulting polynomial be f(x).

So, f(x) = 3x^{3} - 5x^{2} + 6x + k

It is given that when f(x) is divided by (x - 3), the remainder is 8.

Thus, the required number is -46.

### Solution 13

Let the number to be subtracted be k and the resulting polynomial be f(x).

So, f(x) = x^{3} + 3x^{2} - 8x + 14 - k

It is given that when f(x) is divided by (x - 2), the remainder is 10.

Thus, the required number is 8.

### Solution 14

Let f(x) = 2x^{3} - 7x^{2} + ax - 6

x - 2 = 0 x = 2

When f(x) is divided by (x - 2), remainder = f(2)

Let g(x) = x^{3} - 8x^{2} + (2a + 1)x - 16

When g(x) is divided by (x - 2), remainder = g(2)

By the given condition, we have:

f(2) = g(2)

2a - 18 = 4a - 38

4a - 2a = 38 - 18

2a = 20

a = 10

Thus, the value of a is 10.

### Solution 15

## Remainder And Factor Theorems Exercise Ex. 8(B)

### Solution 1(a)

Correct option: (iv) 5

The degree of the polynomial is 5.

Hence, the maximum number of linear factors is 5.

### Solution 1(b)

Correct option: (ii) 16

f(x) = 3x + 8

f(–x) = 3(–x) + 8 = –3x + 8

f(x) + f(–x) = 3x + 8 + (–3x) + 8 = 16

### Solution 1(c)

Correct option: (iii) 0

When x^{25} + x^{24} is divided by (x + 1),

Remainder = (–1)^{25} + (–1)^{24} = –1 + 1 = 0

### Solution 1(d)

Correct option: (ii) x(x – 2)(3x + 4)

3x^{3} – 2x^{2} – 8x

= x(3x^{2} – 2x – 8)

= x(3x^{2} – 6x + 4x – 8)

= x[3x(x – 2) + 4(x – 2)]

= x[(x – 2)(3x + 4)]

= x(x – 2)(3x + 4)

### Solution 1(e)

Correct option: (i) (2 + x)(2 – x)(1 + x)

4 + 4x – x^{2} – x^{3}

= 4(1 + x) – x^{2}(1 + x)

= (4 – x^{2})(1 + x)

= (2 + x)(2 – x)(1 + x)

### Solution 2

(i) Let f(x) = x^{3} - 2x^{2} - 9x + 18

x - 2 = 0 x = 2

Remainder = f(2)

= (2)^{3} - 2(2)^{2} - 9(2) + 18

= 8 - 8 - 18 + 18

= 0

Hence, (x - 2) is a factor of f(x).

Now, we have:

x^{3} - 2x^{2} - 9x + 18 = (x - 2) (x^{2} - 9) = (x - 2) (x + 3) (x - 3)

(ii) Let f(x) = 2x^{3} + 5x^{2} - 28x - 15

x + 5 = 0 x = -5

Remainder = f(-5)

= 2(-5)^{3} + 5(-5)^{2} - 28(-5) - 15

= -250 + 125 + 140 - 15

= -265 + 265

= 0

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

Now, we have:

2x^{3} + 5x^{2} - 28x - 15 = (x + 5) (2x^{2} - 5x - 3)

= (x + 5) [2x^{2} - 6x + x - 3]

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

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

### Solution 3

(i)

(ii) Let f(x) = 2x^{3} + x^{2} - 13x + 6

For x = 2,

f(x) = f(2) = 2(2)^{3} + (2)^{2} - 13(2) + 6 = 16 + 4 - 26 + 6 = 0

Hence, (x - 2) is a factor of f(x).

(iii) f(x) = 3x^{3} + 2x^{2} - 23x - 30

For x = -2,

f(x) = f(-2) = 3(-2)^{3} + 2(-2)^{2} - 23(-2) - 30

= -24 + 8 + 46 - 30 = -54 + 54 = 0

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

(iv) f(x) = 4x^{3} + 7x^{2} - 36x - 63

For x = 3,

f(x) = f(3) = 4(3)^{3} + 7(3)^{2} - 36(3) - 63

= 108 + 63 - 108 - 63 = 0

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

(v) f(x) = x^{3} + x^{2} - 4x - 4

For x = -1,

f(x) = f(-1) = (-1)^{3} + (-1)^{2} - 4(-1) - 4

= -1 + 1 + 4 - 4 = 0

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

### Solution 4

Let f(x) = 3x^{3} + 10x^{2} + x - 6

For x = -1,

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

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

Now, 3x^{3} + 10x^{2} + x - 6 = 0

### Solution 5

f (x) = 2x^{3} - 7x^{2} - 3x + 18

For x = 2,

f(x) = f(2) = 2(2)^{3} - 7(2)^{2} - 3(2) + 18

= 16 - 28 - 6 + 18 = 0

Hence, (x - 2) is a factor of f(x).

### Solution 6

f(x) = x^{3} + 3x^{2} + ax + b

Since, (x - 2) is a factor of f(x), f(2) = 0

(2)^{3} + 3(2)^{2} + a(2) + b = 0

8 + 12 + 2a + b = 0

2a + b + 20 = 0 ...(i)

Since, (x + 1) is a factor of f(x), f(-1) = 0

(-1)^{3} + 3(-1)^{2} + a(-1) + b = 0

-1 + 3 - a + b = 0

-a + b + 2 = 0 ...(ii)

Subtracting (ii) from (i), we get,

3a + 18 = 0

a = -6

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

b = a - 2 = -6 - 2 = -8

f(x) = x^{3} + 3x^{2} - 6x - 8

Now, for x = -1,

f(x) = f(-1) = (-1)^{3} + 3(-1)^{2} - 6(-1) - 8 = -1 + 3 + 6 - 8 = 0

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

### Solution 7

Let f(x) = 4x^{3} - bx^{2} + x - c

It is given that when f(x) is divided by (x + 1), the remainder is 0.

f(-1) = 0

4(-1)^{3} - b(-1)^{2} + (-1) - c = 0

-4 - b - 1 - c = 0

b + c + 5 = 0 ...(i)

It is given that when f(x) is divided by (2x - 3), the remainder is 30.

Multiplying (i) by 4 and subtracting it from (ii), we get,

5b + 40 = 0

b = -8

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

c = -5 + 8 = 3

Therefore, f(x) = 4x^{3} + 8x^{2} + x - 3

Now, for x = -1, we get,

f(x) = f(-1) = 4(-1)^{3} + 8(-1)^{2} + (-1) - 3 = -4 + 8 - 1 - 3 = 0

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

### Solution 8

f(x) = x^{2} + px + q

It is given that (x + a) is a factor of f(x).

g(x) = x^{2} + mx + n

It is given that (x + a) is a factor of g(x).

From (i) and (ii), we get,

pa - q = ma - n

n - q = a(m - p)

Hence, proved.

## Remainder And Factor Theorems Exercise TEST YOURSELF

### Solution 1(a)

Correct option: (ii) –4

x + 1 = 0 ⇒ x = –1

When x^{3} – x^{2} + x – 1 is divided by (x + 1),

Remainder = (–1)^{3} – (–1)^{2} + (–1) – 1

= –1 – 1 – 1 – 1

= –4

### Solution 1(b)

Correct option: (iv) –8

x – 3 = 0 ⇒ x = 3

Since x – 3 is a factor of x^{2} + kx + 15,

Remainder = 0

⇒ (3)^{2} + k(3) + 15 = 0

⇒ 9 + 3k + 15 = 0

⇒ 3k = –24

⇒ k = –8

### Solution 1(c)

Correct option: (i) yes

For (x – 2) to be a factor of x^{3} – 4x^{2} – 11x + 30, remainder = 0

Now,

(2)^{3} – 4(2)^{2} – 11(2) + 30

= 8 – 16 – 22 + 30

= 0

Then, (x – 2) is a factor of given polynomial.

### Solution 1(d)

Correct option: (iii) 7

x – 1 = 0 ⇒ x = 1

Now, 4(1)^{2} – k(1) + 5 = 2

⇒ 4 – k + 5 = 2

⇒ 9 – k = 2

⇒ k = 7

### Solution 1(e)

Correct option: (iv) n – 2m = 4

x – 2 = 0 ⇒ x = 2

Since x – 2 is a factor of mx^{2} – nx + 8,

Remainder = 0

⇒ m(2)^{2} – n(2) + 8 = 0

⇒ 4m – 2n + 8 = 0

⇒ 2m – n + 4 = 0

⇒ n – 2m = 4

### Solution 2

Let f(x) = x^{3} + 3x^{2} - mx + 4

According to the given information,

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

m = 7

### Solution 3

Let the required number be k.

Let f(x) = 3x^{3} - 8x^{2} + 4x - 3 - k

According to the given information,

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

Thus, the required number is -67.

### Solution 4

Let f(x) = x^{3} + (a + 1)x^{2} - (b - 2)x - 6

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

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 ...(i)

Since, (x - 2) is a factor of f(x).

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 ...(ii)

Adding (i) and (ii), we get,

3a - 3 = 0

a = 1

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

1 + b - 8 = 0

b = 7

f(x) = x^{3} + 2x^{2} - 5x - 6

Now, (x + 1) and (x - 2) are factors of f(x). Hence, (x + 1) (x - 2) = x^{2} - x - 2 is a factor of f(x).

f(x) = x^{3} + 2x^{2} - 5x - 6 = (x + 1) (x - 2) (x + 3)

### Solution 5

Let f(x) = x^{2} + ax + b

Since, (x - 2) is a factor of f(x).

Remainder = f(2) = 0

(2)^{2} + a(2) + b = 0

4 + 2a + b = 0

2a + b = -4 ...(i)

It is given that:

a + b = 1 ...(ii)

Subtracting (ii) from (i), we get,

a = -5

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

b = 1 - (-5) = 6

### Solution 6

Let f(x) = mx^{3} + 2x^{2} - 3

g(x) = x^{2} - mx + 4

It is given that f(x) and g(x) leave the same remainder when divided by (x - 2). Therefore, 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

m =

### Solution 7

Let f(x) = px^{3} + 4x^{2} - 3x + q

It is given that f(x) is completely divisible by (x^{2} - 1) = (x + 1)(x - 1).

Therefore, f(1) = 0 and f(-1) = 0

f(1) = p(1)^{3} + 4(1)^{2} - 3(1) + q = 0

p + q + 1 = 0 ...(i)

f(-1) = p(-1)^{3} + 4(-1)^{2} - 3(-1) + q = 0

-p + q + 7 = 0 ...(ii)

Adding (i) and (ii), we get,

2q + 8 = 0

q = -4

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

p = -q - 1 = 4 - 1 = 3

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

Given that f(x) is completely divisible by (x^{2} - 1).

### Solution 8

It is given that when the polynomial x^{3} + 2x^{2} - 5ax - 7 is divided by (x - 1), the remainder is A.

(1)^{3} + 2(1)^{2} - 5a(1) - 7 = A

1 + 2 - 5a - 7 = A

- 5a - 4 = A ...(i)

It is also given that when the polynomial x^{3} + ax^{2} - 12x + 16 is divided by (x + 2), the remainder is B.

x^{3} + ax^{2} - 12x + 16 = B

(-2)^{3} + a(-2)^{2} - 12(-2) + 16 = B

-8 + 4a + 24 + 16 = B

4a + 32 = B ...(ii)

It is also given that 2A + B = 0

Using (i) and (ii), we get,

2(-5a - 4) + 4a + 32 = 0

-10a - 8 + 4a + 32 = 0

-6a + 24 = 0

6a = 24

a = 4

### Solution 9

Let f(x) = (a - 1)x^{3} + (a + 1)x^{2} - (2a + 1)x - 15

It is given that (3x + 5) is a factor of f(x).

f(x) = (a - 1)x^{3} + (a + 1)x^{2} - (2a + 1)x - 15

= 3x^{3} + 5x^{2} - 9x - 15

### Solution 10

Let f(x) = 2x^{3} + 3x^{2} - kx + 5

Using Remainder Theorem, we have

f(2) = 7

∴ 2(2)^{3} + 3(2)^{2} - k(2) + 5 = 7

∴ 16 + 12 - 2k + 5 = 7

∴ 33 - 2k = 7

∴ 2k = 26

∴ k = 13