Remainder & Factor Theorem — Multiple Choice Questions

A polynomial in 'x' is divided by (x - a) and for (x - a) to be a factor of this polynomial, the remainder should be :

1. -a

2. 0

3. a

4. 2a

Answer

For (x - a) to be a factor of a polynomial, the remainder should be equal to zero.

Hence, Option 2 is the correct option.

If p(x) = x + 4, then p(x) + p(-x) = ?

1. 0

2. 8x

3. 8

4. -8

Answer

Given,

⇒ p(x) = x + 4

⇒ p(-x) = -x + 4

⇒ p(x) + p(-x) = x + 4 - x + 4

= 4 + 4

= 8.

Hence, option 3 is the correct option.

If p(x) = x² - $2\sqrt{2}x + 1, p(2\sqrt2)$ = ?

1. 0

2. 1

3. -1

4. -8

Answer

Given,

⇒ p(x) = x² - $2\sqrt{2}x$ + 1

$\Rightarrow p(2\sqrt2) = (2\sqrt2)^2 - (2\sqrt2)(2\sqrt2) + 1 \\[1em] = 2^2 \times 2 - 4 \times 2 + 1 \\[1em] = 8 - 8 + 1 \\[1em] = 1.$

Hence, option 2 is the correct option.

If f(x) = 3x - 5x² - 1, then f(-1) = ?

1. 1

2. -1

3. 7

4. -9

Answer

Given,

⇒ f(x) = 3x - 5x² - 1

⇒ f(-1) = 3(-1) - 5(-1)² - 1

= -3 - 5 - 1

= -9.

Hence, option 4 is the correct option.

If (x¹⁰¹ + 101) is divided by (x + 1), then the remainder is:

1. 102

2. 100

3. 0

4. -101

Answer

The Remainder Theorem states that when a polynomial f(x) is divided by (x - a), the remainder is f(a).

Given,

f(x) = x¹⁰¹ + 101

f(-1) = (-1)¹⁰¹ + 101

= -1 + 101

= 100.

Hence, option 2 is the correct option.

If (3x³ - 5x² + 3x - 7) is divided by (x - 2), then the remainder is :

1. -5

2. 6

3. 3

4. -8

Answer

By remainder theorem,

If a polynomial f(x) is divided by (x - a), remainder = f(a).

Let f(x) = 3x³ - 5x² + 3x - 7

On dividing f(x) by x - 2, remainder = f(2).

f(2) = 3(2)³ - 5(2)² + 3(2) - 7

= 3(8) - 5(4) + 6 - 7

= 24 - 20 + 6 - 7

= 3.

Hence, option 3 is the correct option.

If f(x) = x⁴ - ax³ + x² - ax is divided by (x - a), then the remainder is:

1. 0

2. a

3. -a

4. 2a²

Answer

By remainder theorem,

If a polynomial f(x) is divided by (x - a), remainder = f(a).

Given,

f(x) = x⁴ - ax³ + x² - ax.

On dividing f(x) by x - a, remainder = f(a).

f(a) = a⁴ - a(a)³ + (a)² - a(a)

= a⁴ - a⁴ + a² - a²

= 0.

Hence, option 1 is the correct option.

When f(x) = x³ + ax² + 2x + a is divided by (x + a), then the remainder is :

1. 0

2. -a

3. a

4. 3a

Answer

By remainder theorem,

If a polynomial f(x) is divided by (x - a), remainder = f(a).

Given,

f(x) = x³ + ax² + 2x + a

On dividing f(x) by x + a, remainder = f(-a)

f(-a) = (-a)³ + a(-a)² + 2(-a) + a

= -a³ + a³ - 2a + a

= -a.

Hence, option 2 is the correct option.

If (x² - 7x + a) leaves a remainder 1 when divided by (x + 1), then the value of a is :

1. 1

2. -1

3. -7

4. -5

Answer

Given,

Let f(x) = x² - 7x + a

Given,

On dividing f(x) by x + 1, remainder is 1.

By remainder theorem, remainder = f(-1).

⇒ f(-1) = 1

⇒ (-1)² - 7(-1) + a = 1

⇒ 1 + 7 + a = 1

⇒ a = 1 - 8

⇒ a = -7.

Hence, option 3 is the correct option.

In the division of a cubic polynomial f(x) by a linear polynomial, the remainder is f(-2). Then the divisor must be:

1. (x - 2)

2. (x + 2)

3. (2x + 1)

4. (2x - 1)

Answer

By remainder theorem,

If a polynomial f(x) is divided by (x - a), remainder = f(a).

The remainder is given as f(-2).

⇒ x = -2

⇒ x + 2 = 0.

Hence, option 2 is the correct option.

If p(t) = t² - t - 2, then the value of $p\Big(-\dfrac{1}{3}\Big)$ is :

1. -2

2. $-\dfrac{10}{9}$

3. $-\dfrac{14}{9}$

4. 0

Answer

Given,

p(t) = t² - t - 2

$\Rightarrow p\Big(-\dfrac{1}{3}\Big) = \Big(-\dfrac{1}{3}\Big)^2 - \Big(-\dfrac{1}{3}\Big) - 2 \\[1em] = \Big(\dfrac{1}{9}\Big) + \dfrac{1}{3} - 2 \\[1em] = \dfrac{1 + 1 \times 3 - 2 \times 9}{9} \\[1em] = \dfrac{1 + 3 - 18}{9} \\[1em] = -\dfrac{14}{9}.$

Hence, option 3 is the correct option.

If f(x) = 4x³ - 3x² + 5 is divided by (2x + 1), then the remainder is :

1. $-\dfrac{15}{4}$

2. $-\dfrac{9}{4}$

3. $-\dfrac{5}{4}$

4. $\dfrac{15}{4}$

Answer

Divisor :

⇒ 2x + 1 = 0

⇒ 2x = -1

⇒ x = $-\dfrac{1}{2}$.

Given,

f(x) = 4x³ - 3x² + 5

By remainder theorem,

On dividing f(x) by 2x + 1, remainder = $f\Big(-\dfrac{1}{2}\Big)$.

$\Rightarrow f\Big(-\dfrac{1}{2}\Big) = 4\Big(-\dfrac{1}{2}\Big)^3 - 3\Big(-\dfrac{1}{2}\Big)^2 + 5 \\[1em] = 4\Big(-\dfrac{1}{8}\Big) - 3\Big(\dfrac{1}{4}\Big) + 5 \\[1em] = -\dfrac{1}{2} - \dfrac{3}{4} + 5 \\[1em] = \dfrac{-1 \times 2 - 3 + 5 \times 4}{4} \\[1em] = \dfrac{-2 - 3 + 20}{4} \\[1em] = \dfrac{-5 + 20}{4} \\[1em] = \dfrac{15}{4}.$

Hence, option 4 is the correct option.

If (x⁵⁰ - 1) is divided by (x - 1), then the remainder is :

1. 0

2. -2

3. 49

4. 51

Answer

Let, f(x) = x⁵⁰ - 1.

By remainder theorem,

On dividing f(x) by x - 1, remainder = f(1).

⇒ f(1) = (1)⁵⁰ - 1

= 1 - 1

= 0.

Hence, option 1 is the correct option.

When 2x² - 3kx + k is divided by (x + 2), then the remainder obtained is 5. The value of k is:

1. $-\dfrac{2}{5}$

2. $-\dfrac{3}{5}$

3. $-\dfrac{3}{7}$

4. $-\dfrac{5}{7}$

Answer

Let, f(x) = 2x² - 3kx + k

By remainder theorem,

On dividing f(x) by x + 2, remainder = f(-2).

Given,

Remainder = 5.

⇒ f(-2) = 5

⇒ 2(-2)² - 3k(-2) + k = 5

⇒ 2(4) + 6k + k = 5

⇒ 8 + 7k = 5

⇒ 7k = 5 - 8

⇒ 7k = -3

⇒ k = $-\dfrac{3}{7}$.

Hence, option 3 is the correct option.

If the polynomial p(x) = x³ - 4kx + 3 is divided by (2x + 1), then the remainder obtained is -3. The value of k is:

1. $-\dfrac{17}{16}$

2. $-\dfrac{47}{8}$

3. $-\dfrac{27}{16}$

4. $-\dfrac{47}{16}$

Answer

Let, f(x) = x³ - 4kx + 3

⇒ 2x + 1 = 0

⇒ 2x = -1

⇒ x = $-\dfrac{1}{2}$

By remainder theorem,

On dividing f(x) by 2x + 1, remainder = $f\Big(-\dfrac{1}{2}\Big)$.

Given,

Remainder = -3

$\Rightarrow f\Big(-\dfrac{1}{2}\Big) = -3 \\[1em] \Rightarrow \Big(-\dfrac{1}{2}\Big)^3 - 4k\Big(-\dfrac{1}{2}\Big) + 3 = -3 \\[1em] \Rightarrow \Big(-\dfrac{1}{8}\Big) + \dfrac{4k}{2} = - 3 - 3 \\[1em] \Rightarrow -\dfrac{1}{8} + 2k = - 6 \\[1em] \Rightarrow 2k = -6 + \dfrac{1}{8}\\[1em] \Rightarrow 2k = \dfrac{-6 \times 8 + 1}{8} \\[1em] \Rightarrow k = \dfrac{-48 + 1}{8 \times 2} \\[1em] \Rightarrow k = -\dfrac{47}{16}.$

Hence, option 4 is the correct option.

If (x + 5) is a factor of f(x) = x³ - 20x + 5k, then k = ?

1. -2

2. -3

3. 5

4. -5

Answer

Let, f(x) = x³ - 20x + 5k.

By factor theorem,

If (x + 5) is a factor of f(x), then f(-5) = 0.

⇒ (-5)³ - 20(-5) + 5k = 0

⇒ -125 + 100 + 5k = 0

⇒ -25 + 5k = 0

⇒ 5k = 25

⇒ k = $\dfrac{25}{5}$

⇒ k = 5.

Hence, option 3 is the correct option.

For what value of k is the polynomial f(x) = 2x³ - kx² + 3x + 10 exactly divisible by (x + 2)?

1. 3

2. -3

3. -$\dfrac{1}{3}$

4. -4

Answer

Let, f(x) = 2x³ - kx² + 3x + 10

By the Factor Theorem,

If (x + 2) is a factor of f(x), then f(-2) = 0.

⇒ 2(-2)³ - k(-2)² + 3(-2) + 10 = 0

⇒ 2(-8) - k(4) - 6 + 10 = 0

⇒ -16 - 4k + 4 = 0

⇒ -4k - 12 = 0

⇒ 4k = -12

⇒ k = $-\dfrac{12}{4}$

⇒ k = -3.

Hence, option 2 is the correct option.

If (x⁵⁰ + 2x⁴⁹ + k) is divisible by (x + 1), then the value of k is:

1. 0

2. -1

3. 1

4. -2

Answer

Let, f(x) = x⁵⁰ + 2x⁴⁹ + k

By the Factor Theorem,

If (x + 1) is a factor of f(x), then f(-1) = 0.

⇒ (-1)⁵⁰ + 2(-1)⁴⁹ + k = 0

⇒ 1 + 2(-1) + k = 0

⇒ 1 - 2 + k = 0

⇒ k - 1 = 0

⇒ k = 1.

Hence, option 3 is the correct option.

The value of p for which (x - p) is a factor of x³ - px² + x + 5 is:

1. -5

2. -4

3. 5

4. p + 5

Answer

Given,

f(x) = x³ - px² + x + 5

⇒ x - p = 0

⇒ x = p.

Since, (x - p) is the factor of f(x), thus f(p) = 0.

f(p) = (p)³ - p(p)² + p + 5

⇒ 0 = p³ - p³ + p + 5

⇒ 0 = p + 5

⇒ p = -5.

Hence, option 1 is the correct option.

If x - 2 is a factor of x³ - kx - 12, then the value of k is :

1. 3

2. 2

3. -2

4. -3

Answer

By factor theorem,

If x - a is a factor of f(x), then f(a) = 0.

Given,

x - 2 is a factor of x³ - kx - 12

⇒ x - 2 = 0

⇒ x = 2.

∴ 2³ - 2k - 12 = 0

⇒ 8 - 2k - 12 = 0

⇒ -2k - 4 = 0

⇒ 2k = -4

⇒ k = $-\dfrac{4}{2}$ = -2.

Hence, Option 3 is the correct option.

For a polynomial f(x), f(-1) and f(2) are both equal to zero. Which of the following is a factor of f(x)?

1. x² + x - 2

2. x² - x - 2

3. x² + x + 2

4. x² - 2x + 1

Answer

By the Factor Theorem,

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

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

Since both are the factors of f(x). Multiplying both the factors,

⇒ (x + 1)(x - 2)

⇒ x² - 2x + x - 2

⇒ x² - x - 2.

Hence, option 2 is the correct option.

(x + 1) is a factor of the polynomial :

1. x³ + x² - x + 1

2. x³ + 2x² - x - 2

3. x³ + 4x² - x + 2

4. x³ + x² + 1

Answer

Let, f(x) = x³ + 2x² - x - 2

f(-1) = (-1)³ + 2(-1)² - (-1) - 2

= (-1) + 2(1) + 1 - 2

= -1 + 2 + 1 - 2

= 3 - 3

= 0.

Since f(-1) = 0,

Thus, (x + 1) is factor of x³ + 2x² - x - 2.

Hence, option 2 is the correct option.

A polynomial in x is x³ + 5x² - kx - 24. Which of the following is a factor of the given polynomial so that the value of k is 2?

1. (x + 2)

2. (x - 3)

3. (x + 4)

4. (x - 4)

Answer

By factor theorem,

(x - a) is a factor of f(x), if f(a) = 0.

Given,

Polynomial = x³ + 5x² - kx - 24

If k = 2, then :

Polynomial = x³ + 5x² - 2x - 24.

Dividing polynomial by (x + 4) or substituting -4 in polynomial, we get :

⇒ (-4)³ + 5(-4)² - 2(-4) - 24

⇒ -64 + 5(16) + 8 - 24

⇒ -64 + 80 + 8 - 24

⇒ 88 - 88

⇒ 0.

Since, on substituting -4 in polynomial, we get remainder = 0.

∴ (x + 4) is the factor of x³ + 5x² - kx - 24, when k = 2.

Hence, Option 3 is the correct option.

What number should be subtracted from 2x³ - 5x² + 5x, so that the resulting polynomial has (2x - 3) as a factor?

1. 2

2. -2

3. -3

4. 3

Answer

Let number to be subtracted be a, then resulting polynomial:

f(x) = 2x³ - 5x² + 5x - a,

⇒ 2x - 3 = 0

⇒ 2x = 3

⇒ x = $\dfrac{3}{2}$.

If (2x - 3) is a factor of f(x), then $f\Big(\dfrac{3}{2}\Big) = 0$.

$\Rightarrow 2\Big(\dfrac{3}{2}\Big)^3 - 5\Big(\dfrac{3}{2}\Big)^2 + 5\Big(\dfrac{3}{2}\Big) - a = 0\\[1em] \Rightarrow 2\Big(\dfrac{27}{8}\Big) - 5\Big(\dfrac{9}{4}\Big) + 5\Big(\dfrac{3}{2}\Big) - a = 0\\[1em] \Rightarrow \dfrac{2 \times 27}{8} - \dfrac{5 \times 9}{4} + \dfrac{3 \times 5}{2} -a = 0\\[1em] \Rightarrow \dfrac{27}{4} - \dfrac{45}{4} + \dfrac{15}{2} - a = 0\\[1em] \Rightarrow \dfrac{27 - 45 + 15 \times 2}{4} -a = 0\\[1em] \Rightarrow \dfrac{- 18 + 30}{4} -a = 0 \\[1em] \Rightarrow \dfrac{12}{4} - a = 0 \\[1em] \Rightarrow 3 -a =0\\[1em] \Rightarrow a = 3.$

If 3 is subtracted from 2x³ - 5x² + 5x, then (2x - 3) is a factor.

Hence, option 4 is the correct option.

If (x² + ax + b) is divided by (x + c), then the remainder is :

1. -c² + ac + b

2. c² + ac + b

3. c² - ac - b

4. c² - ac + b

Answer

Given,

f(x) = x² + ax + b

By remainder theorem,

On dividing f(x) by (x + c), remainder = f(-c).

⇒ f(-c) = (-c)² + a(-c) + b

= c² - ac + b.

Hence, option 4 is the correct option.

(x - 2) and (x + 2) are the factors of x³ + x² - 4x - 4. The third factor of the given polynomial is :

1. (x - 1)

2. (x - 4)

3. (x + 1)

4. (x + 4)

Answer

Given,

(x - 2) and (x + 2) are the factors of x³ + x² - 4x - 4.

⇒ (x - 2)(x + 2) = x² - 4

Thus, x² - 4 is also the factor of x³ + x² - 4x - 4.

$\begin{array}{l} \phantom{x - 3x)}{\quad x + 1} \\ x^2 - 4\overline{\smash{\big)}\quad x^3 + x^2 - 4x - 4 } \\ \phantom{x^2 + n}\phantom(\underline{\underset{-}{+}x^3 \underset{+}{-}4x} \\ \phantom{x^2 + 3x -])7 + o)()} x^2 - 4 \\ \phantom{x^2 + 3x - ((5) + )}\underline{\underset{-}{+}x^2 \underset{+}{-}4} \\ \phantom{x^2 + 3x - 5) + 24x )}\times \end{array}$

Thus, x³ + x² - 4x - 4 = (x² - 4)(x + 1)

= (x + 2)(x - 2)(x + 1).

Hence, option 3 is the correct option.

For two polynomials f(x) and g(x), (x - a) and (x - b) are their respective factors. Which of the following is true?

1. f(a) + g(b) = 1

2. f(a) + g(b) = a - b

3. f(a) + g(b) = 0

4. f(a) + g(b) = a + b

Answer

By the Factor Theorem,

If (x - a) is a factor of f(x), then f(a) = 0.

If (x - b) is a factor of g(x), then g(b) = 0.

Thus, f(a) + g(b) = 0.

Hence, option 3 is the correct option.

If the polynomial 2x³ + 3x² - 2x - 3 is completely divisible by (2x + a), and the quotient is equal to (x² - 1), then one of the values of a is :

1. -3

2. -1

3. 1

4. 3

Answer

Given,

The polynomial 2x³ + 3x² - 2x - 3 is completely divisible by (2x + a) and quotient is equal to (x² - 1).

∴ 2x³ + 3x² - 2x - 3 = (2x + a)(x² - 1)

⇒ 2x³ + 3x² - 2x - 3 = 2x³ - 2x + ax² - a

⇒ 2x³ - 2x³ + 3x² - 2x + 2x - 3 = ax² - a

⇒ 3x² - 3 = ax² - a

From above equation,

a = 3.

Hence, Option 4 is the correct option.