Question

How do you use the remainder theorem to evaluate `f(a)=a^4+3a^3-17a^2+2a-7` at a=3?

Answer 1

`f(a)=a^4+3a^3-17a^2+2a-7` at `a=3` is `8`

Explanation:

According to remainder theorem if a polynomial `f(x)` is divided by `(x-p)`, the remainder is `f(p)`.

Hence, using remainder theorem to evaluate `f(a)=a^4+3a^3-17a^2+2a-7` at `a=3`,

we should divide `a^4+3a^3-17a^2+2a-7` by `(a-3)`

this can be done using synthetic division

`3|color(white)(X)1" "color(white)(X)3color(white)(XX)-17" "" "2color(white)(XX)-7`
`color(white)(x)|" "color(white)(Xxx)3color(white)(XXX)18color(white)(Xxxx)3color(white)(Xxxx)15`
`" "stackrel("—————————————----)`
`color(white)(x)|color(white)(X)color(blue)1color(white)(X11)color(red)6color(white)(XXXX)color(red)1color(white)(XXX)color(red)5color(white)(Xxxx)8`

Hence, Quotient is `a^3+6a^2+a+5` and remainder is `8`.

Hence `f(a)=a^4+3a^3-17a^2+2a-7` at `a=3` is `8`