Question

use synthetic division to find the result when $2x^4 - 21x^3 + 8x^2 + 18x + 20$ is divided by $x - 10$. if there is a remainder, express the result in the form $q(x) + \frac{r(x)}{b(x)}$.

Explanation:

Step1: Set up synthetic division

Write the root of $x-10$ (which is 10) and the coefficients of the dividend $2x^4 -21x^3 +8x^2 +18x +20$:

$$\begin{array}{r|rrrrr} 10 & 2 & -21 & 8 & 18 & 20 \\ \hline & & & & & \\ \end{array}$$

Step2: Bring down the leading coefficient

Bring down the 2 to the bottom row:

$$\begin{array}{r|rrrrr} 10 & 2 & -21 & 8 & 18 & 20 \\ \hline & & & & & \\ & 2 & & & & \\ \end{array}$$

Step3: Multiply and add (first column)

Multiply 2 by 10, add to -21:
$2\times10=20$, $-21+20=-1$

$$\begin{array}{r|rrrrr} 10 & 2 & -21 & 8 & 18 & 20 \\ \hline & & 20 & & & \\ & 2 & -1 & & & \\ \end{array}$$

Step4: Multiply and add (second column)

Multiply -1 by 10, add to 8:
$-1\times10=-10$, $8+(-10)=-2$

$$\begin{array}{r|rrrrr} 10 & 2 & -21 & 8 & 18 & 20 \\ \hline & & 20 & -10 & & \\ & 2 & -1 & -2 & & \\ \end{array}$$

Step5: Multiply and add (third column)

Multiply -2 by 10, add to 18:
$-2\times10=-20$, $18+(-20)=-2$

$$\begin{array}{r|rrrrr} 10 & 2 & -21 & 8 & 18 & 20 \\ \hline & & 20 & -10 & -20 & \\ & 2 & -1 & -2 & -2 & \\ \end{array}$$

Step6: Multiply and add (fourth column)

Multiply -2 by 10, add to 20:
$-2\times10=-20$, $20+(-20)=0$

$$\begin{array}{r|rrrrr} 10 & 2 & -21 & 8 & 18 & 20 \\ \hline & & 20 & -10 & -20 & -20 \\ & 2 & -1 & -2 & -2 & 0 \\ \end{array}$$

Step7: Form quotient polynomial

The bottom row (excluding the last number) gives coefficients of $q(x)$: $2x^3 -x^2 -2x -2$, remainder is 0.

Answer:

$2x^3 - x^2 - 2x - 2$