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I got this question which I don't understand:

"Suppose $x^3 - 2x^2 + a = (x + 2) Q(x) + 3$ where $Q(x)$ is a polynomial. Find the value of a."

I know the identity: $P(x)=A(x)Q(x)+R(x)$, but I'm not sure how to apply it in this question.

Any help would be appreciated, thanks.

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2 Answers 2

up vote 1 down vote accepted

Well, when the polynomial is divided by $x+2$ the remainder is $3$, so substitute $-2$ for each $x$ value, and it will equal $3$. Then, you will be able to find $a$.

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The quickest way is to evaluate the identity $x^3 - 2x^2 + a = (x + 2) Q(x) + 3$ for $x = -2$.

You get $-8 - 8 + a = (-2 + 2) Q(-2) + 3$, so $a = 19$,

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