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

Oct 18, 2017

-104 by both means shown below.

Explanation:

Direct Substition

$f \left(- 3\right) = - {\left(- 3\right)}^{4} + {\left(- 3\right)}^{3} - \left(- 3\right) + 1$

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

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 $f \left(- 3\right)$:

$- 3 \rfloor \textcolor{w h i t e}{\text{aaaa")-1color(white)("aaaaa")1color(white)("aaaaa")0color(white)("aaaa")-1color(white)("aaaaaaa}} 1$
$\underline{\textcolor{w h i t e}{\text{aaaaaaaaaaaaaaaa")3color(white)("aaa")-12color(white)("aaaa")36color(white)("aaa}} - 105}$
$\textcolor{w h i t e}{\text{aaaaaaaaa")-1color(white)("aaaa")4color(white)("aaa")-12color(white)("aaaa")35color(white)("aaa}} - 104$

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