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 of the divisor \( x-5=0\Rightarrow x = 5 \). The coefficients of the dividend are \( 1,-1,-17,-15 \).

Step2: Perform synthetic division

Bring down the first coefficient \( 1 \).
Multiply \( 1\times5 = 5 \), add to the second coefficient: \( - 1+5=4 \).
Multiply \( 4\times5 = 20 \), add to the third coefficient: \( -17 + 20=3 \).
Multiply \( 3\times5=15 \), add to the fourth coefficient: \( -15 + 15 = 0 \) (remainder is 0).
The coefficients of the quotient are \( 1,4,3 \) with degree one less than the dividend. So the quotient is \( x^{2}+4x + 3 \).

Answer:

A. \( x^{2}+4x + 3 \)