Mathematics

Problem

The polynomial

g(x)=x^{4}-5x^{3}+6x^{2}+px+q

leaves a remainder of

4

when divided by

(x-1)

and a remainder of

-2

when divided by

(x-2)

. a. (i) Form two equations in

p

and

q

using the given information and hence find the values of

p

and

q

. b. (ii) With the values of

p

and

q

found in part (i), calculate the remainder when

g(x)

is divided by

(x-3)

. c. (iii) For what value of

x

will the remainder obtained in part (ii) become zero? Explain your answer briefly.

Intuitive explanation

This problem uses the Remainder Theorem, which tells you that when a polynomial is divided by $(x-a)$ , the remainder is simply $g(a)$ . Since you know two remainders, you can write two equations by evaluating $g(x)$ at $x=1$ and $x=2$ —this gives you a system to solve for the unknown coefficients $p$ and $q$ . It's a direct way to turn remainder information into algebraic constraints.

Intuitive explanation

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