Question

use synthetic division to simplify \\(\frac{2x^5 + 13x^4 + 5x^3 - 30x^2 - 8x}{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 of the divisor

The divisor is \(x + 2\), so the root is \(x=-2\). The coefficients of the dividend \(2x^{5}+13x^{4}+5x^{3}-30x^{2}-8x\) (note that the constant term is \(0\) since there's no constant term) are \(2, 13, 5, -30, -8, 0\).

Step2: Set up synthetic division

We set up the synthetic division as follows:
\[

$$\begin{array}{r|rrrrrr} -2 & 2 & 13 & 5 & -30 & -8 & 0 \\ & & -4 & -18 & 26 & 8 & 0 \\ \hline & 2 & 9 & -13 & -4 & 0 & 0 \\ \end{array}$$

\]

for each step in synthetic division:

• Bring down the leading coefficient \(2\).
• Multiply \(2\) by \(-2\) to get \(-4\), add to \(13\) to get \(9\).
• Multiply \(9\) by \(-2\) to get \(-18\), add to \(5\) to get \(-13\).
• Multiply \(-13\) by \(-2\) to get \(26\), add to \(-30\) to get \(-4\).
• Multiply \(-4\) by \(-2\) to get \(8\), add to \(-8\) to get \(0\).
• Multiply \(0\) by \(-2\) to get \(0\), add to \(0\) to get \(0\).

Step3: Write the quotient polynomial

The coefficients from the synthetic division give us the quotient polynomial \(q(x)=2x^{4}+9x^{3}-13x^{2}-4x\) and the remainder \(r = 0\). The divisor \(d(x)=x + 2\).

Step4: Write the final form

Since the remainder \(r = 0\), the expression \(\frac{2x^{5}+13x^{4}+5x^{3}-30x^{2}-8x}{x + 2}\) simplifies to \(2x^{4}+9x^{3}-13x^{2}-4x+\frac{0}{x + 2}\), which is just \(2x^{4}+9x^{3}-13x^{2}-4x\) (but in the required form with \(r = 0\) and \(d(x)=x + 2\)).

Answer:

\(2x^{4}+9x^{3}-13x^{2}-4x+\frac{0}{x + 2}\) (or simplified as \(2x^{4}+9x^{3}-13x^{2}-4x\) since the remainder term is zero)