Factorisation — Multiple Choice Questions

When 2x³ - x² - 3x + 5 is divided by 2x + 1, then the remainder is

1. 6
2. -6
3. -3
4. 0

Answer

By remainder theorem, on dividing f(x) by (x - a), the remainder left is f(a).

f(x) = 2x³ - x² - 3x + 5

∴ On dividing f(x) by 2x + 1 or $2(x - \big(-\dfrac{1}{2}\big))$, Remainder = f$\big(-\dfrac{1}{2}\big)$

$f\big(-\dfrac{1}{2}\big) = 2\big(-\dfrac{1}{2}\big)^3 - \big(-\dfrac{1}{2}\big)^2 - 3\big(-\dfrac{1}{2}\big) + 5 \\[1em] = 2\big(-\dfrac{1}{8}\big) - \dfrac{1}{4} + \dfrac{3}{2} + 5 \\[1em] = -\dfrac{1}{4} - \dfrac{1}{4} + \dfrac{3}{2} + 5 \\[1em] = \dfrac{-1 -1 + 6 + 20}{4} \\[1em] = \dfrac{24}{4} \\[1em] = 6$

∴ Option 1, is the correct option.

If on dividing 4x² - 3kx + 5 by x + 2, the remainder is -3 then the value of k is

1. 4
2. -4
3. 3
4. -3

Answer

By remainder theorem, on dividing f(x) by (x - a), the remainder left is f(a).

f(x) = 4x² - 3kx + 5

∴ On dividing f(x) by x + 2 or (x - (-2)), Remainder = f(-2)

Given, remainder = -3

∴ f(-2) = -3

$\Rightarrow 4(-2)^2 - 3k(-2) + 5 = -3 \\[0.5em] \Rightarrow 16 + 6k + 5 = -3 \\[0.5em] \Rightarrow 6k + 21 = -3 \\[0.5em] \Rightarrow 6k = -24 \\[0.5em] k = -4$

∴ Option 2, is the correct option.

If on dividing 2x³ + 6x² - (2k - 7)x + 5 by (x + 3), the remainder is k - 1 then the value of k is

1. 2
2. -2
3. -3
4. 3

Answer

By remainder theorem, on dividing f(x) by (x - a), the remainder left is f(a).

f(x) = 2x³ + 6x² - (2k - 7)x + 5

∴ On dividing f(x) by x + 3 or (x - (-3)), Remainder = f(-3)

Given, remainder = k - 1

∴ f(-3) = k - 1

$\Rightarrow 2(-3)^3 + 6(-3)^2 - (2k - 7)(-3) + 5 = k - 1 \\[0.5em] \Rightarrow -54 + 54 + 6k - 21 + 5 = k - 1 \\[0.5em] \Rightarrow 6k - 16 = k - 1 \\[0.5em] \Rightarrow 6k - k = 16 - 1 \\[0.5em] \Rightarrow 5k = 15 \\[0.5em] \Rightarrow k = 3.$

∴ Option 4, is the correct option.

If x + 1 is a factor of 3x³ + kx² + 7x + 4, then the value of k is

1. -1
2. 0
3. 6
4. 10

Answer

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

f(x) = 3x³ + kx² + 7x + 4

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

∴ f(-1) = 0

$\Rightarrow 3(-1)^3 + k(-1)^2 + 7(-1) + 4 = 0 \\[0.5em] \Rightarrow -3 + k - 7 + 4 = 0 \\[0.5em] \Rightarrow k - 6 = 0 \\[0.5em] k = 6$

∴ Option 3, is the correct option.

What must be subtracted from the polynomial x³ + x² - 2x + 1, so that the result is exactly divisible by (x - 3)?

1. –31

2. –30

3. 30

4. 31

Answer

Polynomial : x³ + x² - 2x + 1

Division by x - 3

⇒ x - 3 = 0

⇒ x = 3.

Let k be subtracted from the polynomial, so resulting polynomial is x³ + x² - 2x + 1 - k.

Resulting polynomial should be exactly divisible by x - 3,

Thus, substituting x = 3, in polynomial x³ + x² - 2x + 1 - k, remainder = 0.

⇒ 3³ + 3² - 2(3) + 1 - k = 0

⇒ 27 + 9 - 6 + 1 - k = 0

⇒ 31 - k = 0

⇒ k = 31.

Hence, Option 4 is the correct option.

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.