Question

How do you evaluate `f(x)=-x^4+x^3-x+1` at x=-3 using direct substitution and synthetic division?

Answer 1

-104 by both means shown below.

Explanation:

Direct Substition

`f(-3) = -(-3)^4 + (-3)^3 - (-3) + 1`

`= -81 - 27 + 3 + 1 = -108 + 3 + 1 = -104`

The potential headaches with direct substituion mainly fall in making sure you properly handle all negatives without making any mistakes.

Synethetic Division

For synthetic division, we use the value -3 as the "outside" value and "divide" by that value following the pattern of synthetic division. The final remainder value (the last value on the bottom row) will be the value of `f(-3)`:

`-3__|color(white)("aaaa")-1color(white)("aaaaa")1color(white)("aaaaa")0color(white)("aaaa")-1color(white)("aaaaaaa")1`
`underline(color(white)("aaaaaaaaaaaaaaaa")3color(white)("aaa")-12color(white)("aaaa")36color(white)("aaa")-105)`
`color(white)("aaaaaaaaa")-1color(white)("aaaa")4color(white)("aaa")-12color(white)("aaaa")35color(white)("aaa")-104`

The potential headaches with synthetic division fall in forgetting to include a 0 coefficient for the missing `x^2` term of the original polynomial.