Question

use synthetic division to find ((x^5 + 2x^3 - 15x - 38) div (x + 2)). 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 the divisor \(x + 2\), the root is \(x=-2\) (since \(x + 2=0\) gives \(x=-2\)). The coefficients of the dividend \(x^{5}+2x^{3}-15x - 38\) are \(1\) (for \(x^{5}\)), \(0\) (for \(x^{4}\)), \(2\) (for \(x^{3}\)), \(0\) (for \(x^{2}\)), \(-15\) (for \(x\)), and \(-38\) (constant term).

Step2: Perform synthetic division

Set up the synthetic division with \(-2\) on the left and the coefficients \(1, 0, 2, 0, -15, -38\) in a row:

\[

$$\begin{array}{r|rrrrrr} -2 & 1 & 0 & 2 & 0 & -15 & -38 \\ & & -2 & 4 & -12 & 24 & -18 \\ \hline & 1 & -2 & 6 & -12 & 9 & -56 \\ \end{array}$$

\]

• Bring down the first coefficient: \(1\).
• Multiply by \(-2\): \(1\times(-2)=-2\). Add to the next coefficient: \(0+(-2)=-2\).
• Multiply \(-2\) by \(-2\): \(-2\times(-2) = 4\). Add to the next coefficient: \(2 + 4=6\).
• Multiply \(6\) by \(-2\): \(6\times(-2)=-12\). Add to the next coefficient: \(0+(-12)=-12\).
• Multiply \(-12\) by \(-2\): \(-12\times(-2)=24\). Add to the next coefficient: \(-15 + 24 = 9\).
• Multiply \(9\) by \(-2\): \(9\times(-2)=-18\). Add to the last coefficient: \(-38+(-18)=-56\).

The last number \(-56\) is the remainder \(r\), and the other numbers are the coefficients of the quotient polynomial \(q(x)\). The quotient polynomial \(q(x)\) has degree \(4\) (since the dividend is degree \(5\)) with coefficients \(1, -2, 6, -12, 9\), so \(q(x)=x^{4}-2x^{3}+6x^{2}-12x + 9\). The divisor \(d(x)=x + 2\).

Step3: Write the result

The division of a polynomial \(f(x)\) by \(d(x)\) is \(f(x)=q(x)d(x)+r\). So \(\frac{x^{5}+2x^{3}-15x - 38}{x + 2}=x^{4}-2x^{3}+6x^{2}-12x + 9+\frac{-56}{x + 2}\) or \(x^{4}-2x^{3}+6x^{2}-12x + 9-\frac{56}{x + 2}\).

Answer:

\(x^{4}-2x^{3}+6x^{2}-12x + 9+\frac{-56}{x + 2}\) (or simplified as \(x^{4}-2x^{3}+6x^{2}-12x + 9-\frac{56}{x + 2}\))