Question

Use the remainder theorem to find P(-3) for P(x) = x^3 +2x²-x-6.Specifically, give the quotient and the remainder for the associated division and the value of P(-3). Quotient=Remainder=P(-3)=

Answer 1

So we're going to use the remainder theorem.

Remember that the remainder theorem is stated as follows: When a polynomial p(x) is divided by a linear polynomial b(x) whose zero is x = k, the remainder is given by r = p(k). This is, in our problem:

`\begin{gathered} P(x)=x^3+2x^2-x-6 \\ r=P(-3) \\ b(x)=x+3 \end{gathered}`

The first thing we're going to do is to divide:

`(x^3+2x^2-x-6)/(x+3)`

Using synthetic divition:

And, the value of P(-3) is -12.

So,

`\begin{gathered} \text{Quotient = }x^2-x+2 \\ \text{Remainder}=-12 \\ P(-3)=-12 \end{gathered}`

Using the theorem, notice that the remainder and P(-3) are the same.