Mathematics

Problem

For the polynomial

p(x)=x^{3}+ax^{2}+bx+6

, it is known that

x=1

is a factor and the remainder when

p(x)

is divided by

x-2

10

. Which ordered pair

(a,b)

satisfies these conditions? a)

(0,-7)

(-5,-2)

(5,-12)

(8,-15)

Intuitive explanation

The key is to translate the two given conditions into equations using the Factor Theorem and Remainder Theorem. Since $x=1$ is a factor, $p(1)=0$ ; since dividing by $x-2$ leaves remainder $10$ , we have $p(2)=10$ . These two equations in $a$ and $b$ are all you need to solve the system and find the unique pair.

Intuitive explanation

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