Question

use synthetic division to find ((x^{4} + 8x^{3} - 13x - 6) div (x + 1)).
write your answer in the form (q(x) + \frac{r}{d(x)}), where (q(x)) is a polynomial, (r) is an integer, and (d(x)) is a linear polynomial. simplify any fractions.

Explanation:

Step1: Identify the root for synthetic division

For the divisor \(x + 1\), the root \(c\) is \(-1\) (since \(x+1 = 0\) gives \(x=-1\)). The polynomial \(x^{4}+8x^{3}-13x - 6\) has coefficients: for \(x^{4}\): \(1\), \(x^{3}\): \(8\), \(x^{2}\): \(0\) (since there is no \(x^{2}\) term), \(x\): \(-13\), constant term: \(-6\).

Step2: Set up synthetic division

Write the coefficients: \(1\), \(8\), \(0\), \(-13\), \(-6\) and the root \(-1\) on the left.

Step3: Perform synthetic division

• Bring down the first coefficient: \(1\).
• Multiply by \(-1\): \(1\times(-1)=-1\). Add to the next coefficient: \(8 + (-1)=7\).
• Multiply \(7\) by \(-1\): \(7\times(-1)=-7\). Add to the next coefficient: \(0+(-7)=-7\).
• Multiply \(-7\) by \(-1\): \(-7\times(-1) = 7\). Add to the next coefficient: \(-13 + 7=-6\).
• Multiply \(-6\) by \(-1\): \(-6\times(-1)=6\). Add to the last coefficient: \(-6 + 6 = 0\).

The coefficients of the quotient polynomial \(q(x)\) are \(1\), \(7\), \(-7\), \(-6\) (degree one less than the original polynomial, so degree \(3\)) and the remainder \(r = 0\).

So \(q(x)=x^{3}+7x^{2}-7x - 6\), \(r = 0\), \(d(x)=x + 1\).

Answer:

\(x^{3}+7x^{2}-7x - 6+\frac{0}{x + 1}\) (or simply \(x^{3}+7x^{2}-7x - 6\) since the remainder term is zero)