Question

if $g(x) = 7x^3 + 32x^2 - 19x - 20$, use synthetic division to find $g(-5)$.

Explanation:

Step1: Set up synthetic division

We use the root \( x = -5 \) and the coefficients of \( g(x) = 7x^3 + 32x^2 - 19x - 20 \), which are \( 7, 32, -19, -20 \).
Set up the synthetic division as:
\[

$$\begin{array}{r|rrrr} -5 & 7 & 32 & -19 & -20 \\ & & -35 & 15 & 20 \\ \hline & 7 & -3 & -4 & 0 \\ \end{array}$$

\]

Step2: Interpret the result

In synthetic division, the last number in the bottom row is the remainder, which is equal to \( g(-5) \) by the Remainder Theorem. Here, the last number is \( 0 \).

Answer:

\( 0 \)