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 substitution

Write the coefficients of the polynomial: \(-4\), \(9\), \(8\), \(9\), \(-13\) and the value \(x = 3\) for synthetic division.
\[

$$\begin{array}{r|rrrrr} 3 & -4 & 9 & 8 & 9 & -13 \\ & & & & & \\ \hline & -4 & & & & \\ \end{array}$$

\]

Step2: Bring down the leading coefficient

Bring down the \(-4\).
\[

$$\begin{array}{r|rrrrr} 3 & -4 & 9 & 8 & 9 & -13 \\ & & & & & \\ \hline & -4 & & & & \\ \end{array}$$

\]

Step3: Multiply and add for each column

• Multiply \(-4\) by \(3\) to get \(-12\), then add to \(9\): \(9 + (-12) = -3\)

\[

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

\]

• Multiply \(-3\) by \(3\) to get \(-9\), then add to \(8\): \(8 + (-9) = -1\)

\[

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

\]

• Multiply \(-1\) by \(3\) to get \(-3\), then add to \(9\): \(9 + (-3) = 6\)

\[

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

\]

• Multiply \(6\) by \(3\) to get \(18\), then add to \(-13\): \(-13 + 18 = 5\)

\[

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

\]

Answer:

The remainder is \(5\), so the value of \(f(3)\) is \(5\).