Question

use synthetic substitution to evaluate the given value and determine the remainder. f (x) = -4x⁴ + 9x³ + 8x² + 9x - 13 at x = 3

Explanation:

Step1: Set up synthetic division

The coefficients of \( f(x)=-4x^{4}+9x^{3}+8x^{2}+9x - 13 \) are \(-4, 9, 8, 9, -13\), and we use \( x = 3 \) for synthetic substitution.
\[

$$\begin{array}{r|rrrrr} 3 & -4 & 9 & 8 & 9 & -13 \\ & & -12 & -9 & -3 & 18 \\ \hline & -4 & -3 & -1 & 6 & 5 \\ \end{array}$$

\]

Step2: Interpret the result

In synthetic substitution, the last number in the bottom row is the remainder (and also the value of \( f(3) \)).

Answer:

The remainder when \( f(x)=-4x^{4}+9x^{3}+8x^{2}+9x - 13 \) is evaluated at \( x = 3 \) using synthetic substitution is \( 5 \).