Question

use synthetic division to simplify \\(\frac{9x^2 - 30}{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 of the divisor

The divisor is \(x + 1\), so the root is \(x=-1\) (since \(x + 1=0\) gives \(x=-1\)). The dividend is \(9x^{2}+0x - 30\) (we add the missing \(x\) term with coefficient 0).

Step2: Set up synthetic division

We write the coefficients of the dividend: 9 (for \(x^{2}\)), 0 (for \(x\)), -30, and the root -1 on the left.
\[

$$\begin{array}{r|rrr} -1 & 9 & 0 & -30 \\ & & -9 & 9 \\ \hline & 9 & -9 & -21 \\ \end{array}$$

\]

• Bring down the 9.
• Multiply -1 by 9 to get -9, add to 0: \(0+(-9)=-9\).
• Multiply -1 by -9 to get 9, add to -30: \(-30 + 9=-21\).

Step3: Interpret the result

The quotient polynomial \(q(x)\) has coefficients 9 (for \(x\)) and -9, so \(q(x)=9x - 9\). The remainder \(r=-21\) and the divisor \(d(x)=x + 1\).

Answer:

\(9x - 9+\frac{-21}{x + 1}\) (or \(9x - 9-\frac{21}{x + 1}\))