Question

use synthetic division to solve \\(\left(x^{3}-x^{2}-17x - 15\
ight)\div(x - 5)\\). what is the quotient?\
\\(\circ\\) \\(x^{2}+4x + 3\\)\
\\(\circ\\) \\(x^{2}-6x + 13-\frac{80}{x - 5}\\)\
\\(\circ\\) \\(x^{3}+4x^{2}+3x\\)\
\\(\circ\\) \\(x^{2}-6x + 13-\frac{80}{x + 5}\\)

Explanation:

Step1: Set up synthetic division

For dividing \(x^3 - x^2 - 17x - 15\) by \(x - 5\), we use the root \(r = 5\) (since \(x - 5 = 0\) gives \(x = 5\)). The coefficients of the dividend are \(1\) (for \(x^3\)), \(-1\) (for \(x^2\)), \(-17\) (for \(x\)), and \(-15\) (constant term).
Set up the synthetic division as:

$$
LATEXBLOCK0
$$

Step2: Interpret the results

The numbers in the bottom row (excluding the last one) are the coefficients of the quotient polynomial. The last number is the remainder (which is \(0\) here).
The degree of the quotient is one less than the dividend (since we divided by a linear term). So the quotient is \(x^2 + 4x + 3\) (since the coefficients are \(1\) for \(x^2\), \(4\) for \(x\), and \(3\) for the constant term).

Answer:

\(x^2 + 4x + 3\) (corresponding to the first option: \(x^2 + 4x + 3\))