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Number Theory : Division of Polynomial

Remainders of division of a polynomial

Question

What number should be subtracted from x³ + 4x² - 7x + 12 if it is to be perfectly divisible by x + 3?

(1) 42
(2) 39
(3) 13
(4) None of these

Correct Choice is (1) and Correct Answer is 42

Explanatory Answer

According to remainder theorem when , then the remainder is f(-a).

In this case, as x + 3 divides x³ + 4x² - 7x + 12 - k perfectly (k being the number to be subtracted), the remainder is 0 when the value of x is substituted by -3.

i.e., (-3)³ + 4(-3)² - 7(-3) + 12 - k = 0

or -27 + 36 + 21 + 12 = k

or k = 42

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