Question

How do you use the remainder theorem to evaluate `f(a)=a^3+3a^2+2a+8` at a=-3?

Answer 1

`f(-3)=2`

Explanation:

`f(a)=a^3+3a^2+2a+8` at `a=-3`

Use synthetic division.

`ul(-3)|1 color(white)(aaaaa)3color(white)(aaa)2color(white)(aaaa)8`
`color(white)(aaaa)ul{darrcolor(white)(aa)-3color(white)(aaa)0color(white)(a)-6`
`color(white)(a^2aaa)1color(white)(aaaaa)0color(white)(aaa)2color(white)(aa^2a)color(red)2`

The last number `color(red)2` is the remainder and is also the
value of `f(-3).`

`f(-3)=color(red)2`

You can check your solution by substituting `x=-3`.

`f(-3)=(-3)^3+3(-3)^2+2(-3)+8=2`