Question

use synthetic division and the remainder theorem to find p(a).
p(x) = x³ + 5x² - 8x + 2; a = 4
p(a) =
(simplify your answer.)

Explanation:

Step1: Set up synthetic division

Write coefficients of $P(x)$ and $a=4$:

$$\begin{array}{r|rrrr} 4 & 1 & 5 & -8 & 2 \\ \hline & & & & \end{array}$$

Step2: Bring down leading coefficient

Carry 1 to the bottom row:

$$\begin{array}{r|rrrr} 4 & 1 & 5 & -8 & 2 \\ \hline & & & & \\ \hline & 1 & & & \end{array}$$

Step3: Multiply and add (column 2)

Multiply 1 by 4, add to 5:
$1 \times 4 = 4$, $5 + 4 = 9$

$$\begin{array}{r|rrrr} 4 & 1 & 5 & -8 & 2 \\ \hline & & 4 & & \\ \hline & 1 & 9 & & \end{array}$$

Step4: Multiply and add (column3)

Multiply 9 by 4, add to -8:
$9 \times 4 = 36$, $-8 + 36 = 28$

$$\begin{array}{r|rrrr} 4 & 1 & 5 & -8 & 2 \\ \hline & & 4 & 36 & \\ \hline & 1 & 9 & 28 & \end{array}$$

Step5: Multiply and add (column4)

Multiply 28 by 4, add to 2:
$28 \times 4 = 112$, $2 + 112 = 114$

$$\begin{array}{r|rrrr} 4 & 1 & 5 & -8 & 2 \\ \hline & & 4 & 36 & 112 \\ \hline & 1 & 9 & 28 & 114 \end{array}$$

By Remainder Theorem, the final value is $P(4)$.

Answer:

114