Question

How do you evaluate `f(x)=-3x^3+7x^2-4x+8` at x=3 using direct substitution and synthetic division?

Answer 1

The remainder is `color(red)(-22)` and the quotient is `=-3x^2-2x-10`

Explanation:

Let's perform the synthetic division

`color(white)(aaaa)` `3` `color(white)(aaaaaa)` `|` `color(white)(aa)` `-3` `color(white)(aaaaaaa)` `7` `color(white)(aaaaaa)` `-4` `color(white)(aaaaaaa)` `8`
`color(white)(aaaaaaaaaaaa)` `------------`

`color(white)(aaaa)` `color(white)(aaaaaaa)` `|` `color(white)(aaaa)` `color(white)(aaaaaa)` `-9` `color(white)(aaaaaa)` `-6` `color(white)(aaaaa)` `-30`
`color(white)(aaaaaaaaaaaa)` `------------`

`color(white)(aaaa)` `color(white)(aaaaaaa)` `|` `color(white)(aaa)` `-3` `color(white)(aaaaa)` `-2` `color(white)(aaaaa)` `-10` `color(white)(aaaaa)` `color(red)(-22)`

The remainder is `color(red)(-22)` and the quotient is `=-3x^2-2x-10`

Therefore,

`(-3x^3+7x^2-4x+8)=(-3x^2-2x-10)-22/(x-3)`

Also,

`f(3)=-3*(3)^3+7*(3)^2-4*(3)*8=-81+63-12+8=-93+71=-22`