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On dividing $3 x 3 + x 2 + 2 x + 5$ by a polynomial g(x), the quotient and remainder are (3x – 5) and (9x + 10) respectively, then the value of g(x) is ____.

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Detailed Solution

On dividing

3 x 3 + x 2 + 2 x + 5

by a polynomial g(x), the quotient and remainder are (3x – 5) and (9x + 10) respectively, then the value of g(x) is

x 2 + 2 x + 1

.
Now, if a polynomial say

f x

is divided by another polynomial say

g (x)

to give the quotient as say

q x

and the remainder as

r x

, then by the division algorithm for polynomials, we have,

f (x) = q (x) × g (x) + r (x) .......... (i)

The given values of the dividend, quotient and remainder are as follows,

f x = 3 x 3 + x 2 + 2 x + 5 q x = 3 x − 5 r x = 9 x + 10

Substituting these values in the equation (i) and simplifying, we get,

(3 x − 5) × g (x) + (9 x + 10) = 3 x 3 + x 2 + 2 x + 5 (3 x − 5) × g (x) = 3 x 3 + x 2 + 2 x + 5 − (9 x + 10) (3 x − 5) × g (x) = 3 x 3 + x 2 − 7 x − 5 g (x) = 3 x 3 + x 2 − 7 x − 5 (3 x − 5)

So, we will find the value of

g (x)

by performing long division as follows,
Question Image

So, the quotient obtained after the long division is

x 2 + 2 x + 1

.
Hence, the value of

g (x)

x 2 + 2 x + 1

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