Question

factoring higher-order polynomials
$4x^4 - 8x^3 - 63x^2 + 45x + 162$
the root is: 2
rewrite the root as a linear expression with integer constants:
is the divisor given in $(x - r)$ form?
how many terms are in the dividend?
synthetic division table

Explanation:

Step1: Set up synthetic division

Write root $r=2$, dividend coefficients $4, -8, -63, 45, 162$.

$$\begin{array}{c|ccccc} 2 & 4 & -8 & -63 & 45 & 162 \\ \hline & & & & & \\ \hline & 4 & & & & \\ \end{array}$$

Step2: Multiply and add (1st term)

Multiply $4 \times 2 = 8$, add to next coefficient: $-8 + 8 = 0$.

$$\begin{array}{c|ccccc} 2 & 4 & -8 & -63 & 45 & 162 \\ \hline & & 8 & & & \\ \hline & 4 & 0 & & & \\ \end{array}$$

Step3: Multiply and add (2nd term)

Multiply $0 \times 2 = 0$, add to next coefficient: $-63 + 0 = -63$.

$$\begin{array}{c|ccccc} 2 & 4 & -8 & -63 & 45 & 162 \\ \hline & & 8 & 0 & & \\ \hline & 4 & 0 & -63 & & \\ \end{array}$$

Step4: Multiply and add (3rd term)

Multiply $-63 \times 2 = -126$, add to next coefficient: $45 + (-126) = -81$.

$$\begin{array}{c|ccccc} 2 & 4 & -8 & -63 & 45 & 162 \\ \hline & & 8 & 0 & -126 & \\ \hline & 4 & 0 & -63 & -81 & \\ \end{array}$$

Step5: Multiply and add (4th term)

Multiply $-81 \times 2 = -162$, add to last coefficient: $162 + (-162) = 0$.

$$\begin{array}{c|ccccc} 2 & 4 & -8 & -63 & 45 & 162 \\ \hline & & 8 & 0 & -126 & -162 \\ \hline & 4 & 0 & -63 & -81 & 0 \\ \end{array}$$

Answer:

Rewrite the root as a linear expression: $x-2$
Is the divisor in $(x-r)$ form? Yes
How many terms are in the dividend? 5
Synthetic Division Table (middle column, top to bottom): $8, 0, -126, -162$
Synthetic Division Table (left column, top to bottom): $4, 0, -63, -81, 0$