Question

How do you evaluate `f(x)=x^3+3x^2-2x+5` at x=-3 using direct substitution and synthetic division?

Answer 1

Quotient `color(blue)(Q = x^2 - 2)`, Remainder `color(blue)(R = 11 / (x + 3))`

Explanation:

Given equation is

`f(x) = x^3 + 3x^2 - 2x + 5` at x = -3

Substitution

`f(x) = f(3) = -3^3 + (3 * -3^2) - (2*-3) + 5 = 53`

Synthetic Division

`color(white)(aa)-3color(white)(aa)|color(white)(aa)1color(white)(aaaa)3color(white)(aa)-2color(white)(aa)5`
`color(white)(aaaaaaa)|color(white)(aa)darrcolor(white)(a)-3color(white)(aaaa)0color(white)(aa)6`
`color(white)(aaaaaaa)----------`
`color(white)(aaaaaaaaaaaa)1color(white)(aaa)0color(white)(aa)-2color(white)(aa)11`

Quotient `Q = x^2 - 2`, Remainder `R = 11 / (x + 3)`