Question

use the long division method to find the result when $x^3 + 5x^2 - 5x - 8$ is divided by $x + 1$. if there is a remainder, express the result in the form $q(x) + \frac{r(x)}{b(x)}$.

Explanation:

Step1: Divide the leading terms

Divide the leading term of the dividend \(x^3\) by the leading term of the divisor \(x\), we get \(x^2\). Multiply the divisor \(x + 1\) by \(x^2\) to get \(x^3+x^2\). Subtract this from the dividend:

$$
LATEXBLOCK0
$$

Step2: Divide the new leading terms

Divide the leading term of \(4x^2-5x - 8\) (which is \(4x^2\)) by the leading term of the divisor \(x\), we get \(4x\). Multiply the divisor \(x + 1\) by \(4x\) to get \(4x^2+4x\). Subtract this from \(4x^2-5x - 8\):

$$
LATEXBLOCK1
$$

Step3: Divide the new leading terms

Divide the leading term of \(-9x - 8\) (which is \(-9x\)) by the leading term of the divisor \(x\), we get \(-9\). Multiply the divisor \(x + 1\) by \(-9\) to get \(-9x-9\). Subtract this from \(-9x - 8\):

$$
LATEXBLOCK2
$$

Step4: Write the result

The quotient \(q(x)\) is \(x^2 + 4x-9\) and the remainder \(r(x)\) is \(1\), and the divisor \(b(x)\) is \(x + 1\). So the result is \(x^2 + 4x-9+\frac{1}{x + 1}\).

Answer:

\(x^2 + 4x - 9+\frac{1}{x + 1}\)