Question

use synthetic division to find the remainder.\\(\dfrac{x^{4}+3x^{3}+x^{2}-3x + 3}{x + 2}\\)

Explanation:

Step1: Identify the root of the divisor

For the divisor \(x + 2\), set \(x+2 = 0\), so \(x=-2\). We will use \(-2\) in synthetic division. The coefficients of the dividend \(x^{4}+3x^{3}+x^{2}-3x + 3\) are \(1,3,1,-3,3\) (corresponding to \(x^{4},x^{3},x^{2},x, \text{constant term}\)).

Step2: Set up synthetic division

Write the root \(-2\) on the left and the coefficients \(1,3,1,-3,3\) in a row:
\[

$$\begin{array}{r|rrrrr} -2 & 1 & 3 & 1 & -3 & 3 \\ & & -2 & -2 & 2 & 2 \\ \hline & 1 & 1 & -1 & -1 & 5 \\ \end{array}$$

\]

• Bring down the first coefficient \(1\).
• Multiply \(-2\) by \(1\) to get \(-2\), add to the next coefficient \(3\): \(3+(-2)=1\).
• Multiply \(-2\) by \(1\) to get \(-2\), add to the next coefficient \(1\): \(1+(-2)= - 1\).
• Multiply \(-2\) by \(-1\) to get \(2\), add to the next coefficient \(-3\): \(-3 + 2=-1\).
• Multiply \(-2\) by \(-1\) to get \(2\), add to the last coefficient \(3\): \(3+2 = 5\).

The last number in the bottom row is the remainder.

Answer:

The remainder is \(5\).