Question

use synthetic division to find the result when $4x^4 + 2x^3 - 2x^2 + 23x - 2$ is divided by $x + 2$. if there is a remainder, express the result in the form $q(x) + \frac{r(x)}{b(x)}$.

Explanation:

Step1: Identify the root

For the divisor \(x + 2\), set \(x+2 = 0\), so \(x=-2\). We will use \(-2\) in synthetic division.

Step2: List the coefficients

The polynomial is \(4x^{4}+2x^{3}-2x^{2}+23x - 2\), so the coefficients are \(4, 2, -2, 23, -2\).

Step3: Perform synthetic division

Bring down the first coefficient (\(4\)):

$$
LATEXBLOCK0
$$

Multiply \(4\) by \(-2\) to get \(-8\), add to the next coefficient (\(2\)): \(2+(-8)= - 6\)

$$
LATEXBLOCK1
$$

Multiply \(-6\) by \(-2\) to get \(12\), add to the next coefficient (\(-2\)): \(-2 + 12=10\)

$$
LATEXBLOCK2
$$

Multiply \(10\) by \(-2\) to get \(-20\), add to the next coefficient (\(23\)): \(23+(-20) = 3\)

$$
LATEXBLOCK3
$$

Multiply \(3\) by \(-2\) to get \(-6\), add to the last coefficient (\(-2\)): \(-2+(-6)=-8\)

$$
LATEXBLOCK4
$$

The last number (\(-8\)) is the remainder. The other numbers are the coefficients of the quotient polynomial, starting from degree \(3\) (since we divided a degree \(4\) polynomial by a degree \(1\) polynomial). So the quotient \(q(x)=4x^{3}-6x^{2}+10x + 3\) and the remainder \(r(x)=-8\), and the divisor \(b(x)=x + 2\).

Answer:

\(4x^{3}-6x^{2}+10x + 3+\frac{-8}{x + 2}\) (or \(4x^{3}-6x^{2}+10x + 3-\frac{8}{x + 2}\))