Question

use synthetic division to simplify \\(\frac{x^5 - 29x^3 + 26x^2 - 27x - 15}{x - 5}\\). 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 \(x^{5}-29x^{3}+26x^{2}-27x - 15\) has coefficients: for \(x^{5}\): 1, \(x^{4}\): 0 (since there's no \(x^{4}\) term), \(x^{3}\): -29, \(x^{2}\): 26, \(x\): -27, constant: -15. The root for \(x - 5\) is \(x = 5\).

Step2: Set up synthetic division

Write the root (5) and the coefficients: 1, 0, -29, 26, -27, -15.
Bring down the first coefficient (1):
\[

$$\begin{array}{r|rrrrrr} 5 & 1 & 0 & -29 & 26 & -27 & -15 \\ & & 5 & 25 & -20 & 30 & 15 \\ \hline & 1 & 5 & -4 & 6 & 3 & 0 \\ \end{array}$$

\]
Multiply 1 by 5 = 5, add to next coefficient (0 + 5 = 5). Multiply 5 by 5 = 25, add to -29 (-29 + 25 = -4). Multiply -4 by 5 = -20, add to 26 (26 - 20 = 6). Multiply 6 by 5 = 30, add to -27 (-27 + 30 = 3). Multiply 3 by 5 = 15, add to -15 (-15 + 15 = 0).

Step3: Form the quotient polynomial

The coefficients of the quotient polynomial (from synthetic division) are 1 ( \(x^{4}\) ), 5 ( \(x^{3}\) ), -4 ( \(x^{2}\) ), 6 ( \(x\) ), 3 (constant). So \(q(x)=x^{4}+5x^{3}-4x^{2}+6x + 3\) and the remainder \(r = 0\) (since the last number is 0) and \(d(x)=x - 5\).

Answer:

\(x^{4}+5x^{3}-4x^{2}+6x + 3+\frac{0}{x - 5}\) (or simply \(x^{4}+5x^{3}-4x^{2}+6x + 3\) since the remainder term is 0)