Question

How do you use synthetic substitution to evaluate f(-3) for `f(x)=x^4-4x^3+2x^2-4x+6`?

Answer 1

`f(-3) = 225`

Explanation:

Given: `f(x) = x^4 - 4x^3 +2x^2 - 4x + 6`

`f(-3) = (-3)^4 - 4(-3)^3 + 2(-3)^2 - 4(-3) +6 = 225`

According to the Remainder Theorem: `(f(x))/(x- -3) = (f(x))/(x+3) = 225`

Synthetic Division , where `x+3 = 0" or " x = -3:`

terms:`" "x^4" "x^3" "x^2" "x" constant"`

`ul(-3)| " "1" "-4" "2" "-4" "6`
`" "ul(+" "-3" "21" "-69" "219)`
`" "1" "-7" "23" "-73" "225`

terms:`" "x^3" "x^2" "x" constant, remainder"`

This means

`(x^4-4x^3+2x^2-4x+6)/(x+3) = x^3 - 7x^2 +23x -73 +225/(x+3)`

The remainder of synthetic division (the last value) is `f(-3) = 225`