Question

question #8
troy is attempting to determine if x + 7 is a factor of x³+4x² - 19x + 14. his work is shown below. determine if troy made a mistake.
7 | 1 4 -19 14
| 7 77 406
|__________
| 1 11 58 420
troys conclusion: x + 7 is not a factor of x³+4x² - 19x + 14 because the remainder is not 0.
troy did not make a mistake.
troy made an arithmetic mistake doing the synthetic division.
troys conclusion is wrong. x + 7 is a factor because the remainder is not 0.
troys mistake was that he used 7 instead of -7 when doing his synthetic division.

Explanation:

Step1: Recall factor - theorem rule

For $x - a$ to be a factor of a polynomial $P(x)$, $P(a)=0$. When using synthetic division to check if $x + 7$ is a factor, we should use $a=-7$ (since $x+7=x - (-7)$) in synthetic division.

Step2: Analyze Troy's work

Troy used 7 in the synthetic - division setup instead of - 7.

Answer:

Troy's mistake was that he used 7 instead of - 7 when doing his synthetic division.