Question

use synthetic division to find ((x^3 + x - 20) 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 coefficients and root

The polynomial \(x^3 + x - 20\) can be written as \(x^3 + 0x^2 + x - 20\). The divisor is \(x + 2\), so the root \(c=-2\) (since \(x+2 = x - (-2)\)). The coefficients of the dividend are \(1\) (for \(x^3\)), \(0\) (for \(x^2\)), \(1\) (for \(x\)), and \(-20\) (constant term).

Step2: Set up synthetic division

Write the root \(-2\) to the left, and the coefficients \(1, 0, 1, -20\) in a row:
\[

$$\begin{array}{r|rrrr} -2 & 1 & 0 & 1 & -20 \\ & & -2 & 4 & -10 \\ \hline & 1 & -2 & 5 & -30 \\ \end{array}$$

\]

• Bring down the first coefficient (\(1\)).
• Multiply \(-2\) by \(1\) to get \(-2\), add to the next coefficient (\(0 + (-2)= -2\)).
• Multiply \(-2\) by \(-2\) to get \(4\), add to the next coefficient (\(1 + 4 = 5\)).
• Multiply \(-2\) by \(5\) to get \(-10\), add to the last coefficient (\(-20 + (-10)= -30\)).

Step3: Interpret the result

The numbers in the bottom row (excluding the last one) are the coefficients of the quotient polynomial \(q(x)\), and the last number is the remainder \(r\). So, \(q(x)=x^2 - 2x + 5\), \(r = -30\), and \(d(x)=x + 2\).

Answer:

\(x^2 - 2x + 5 + \frac{-30}{x + 2}\) (or simplified as \(x^2 - 2x + 5 - \frac{30}{x + 2}\))