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The polynomial $x 3 − 10 x 2 + a x + b$ is exactly divisible by $x − 1$ as well as $x − 2$ . The values of a and b are

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a = 22, b = 14

a = 23, b = 14

a= 23, b = -14

a = 21, b = -14

answer is C.

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Detailed Solution

Given that, a polynomial is

x 3 − 10 x 2 + a x + b

.
By factor theorem, if ( x - a) is a factor of f(x), then f(a) = 0.
Let us assume that

f x = x 3 − 10 x 2 + a x + b

.
Calculating the value of x using

x − 1 = 0

and

x − 2 = 0

, we get,

⇒ (x − 1) = 0 ⇒ x = 1

And,

(x − 2) = 0 x = 2

Therefore, values of x are 1 and 2.
Applying the factor theorem and substituting the value of x , we infer that

f x

is exactly divisible by

x − 1

and

x − 2

.
So,

x − 1

and

x − 2

are factors of

f x

.
According to the factor theorem,

x − 1

and

x − 2

will be factors of

f x

when

f 1 = 0 f 2 = 0

Applying these conditions, we get,

f 1 = 0

Calculating the value of ‘a’ and ‘b’ by simplifying (1) and (2), we get

⇒ a + b = 9 ⇒ 2 a + 2 b = 18 … … (3)

Subtracting (2) from (3), we get,

⇒ (2 a + 2 b) − (2 a + b) = 18 − 32 ⇒ b = − 14

Substituting the value of ‘b’ in (1) and calculating the value of ‘a’, we get,

⇒ a + b = 9 ⇒ a = 9 − b ⇒ a = 9 − (− 14) ⇒ a = 23

The values of a and b are 23 and -14 respectively.
Hence, option 3 is correct.

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