Question

which statement is true of $x^2 + 8x - 6$?\\(\bigcirc\\) $(x - 2)$ is a factor of the polynomial.\\(\bigcirc\\) $(x + 2)$ is a factor of the polynomial.\\(\bigcirc\\) $(x - 4)$ is a factor of the polynomial.\\(\bigcirc\\) the polynomial is prime.

Explanation:

Step1: Use Factor Theorem (x-k) test

For a factor $(x-k)$, evaluate $f(k)=0$.

Test (x-2):

$f(2) = 2^2 + 8(2) -6 = 4 +16 -6 =14
eq0$

Test (x+2):

$f(-2) = (-2)^2 +8(-2)-6=4-16-6=-18
eq0$

Test (x-4):

$f(4)=4^2+8(4)-6=16+32-6=42
eq0$

Step2: Check discriminant for primality

Discriminant formula: $\Delta = b^2-4ac$
$\Delta = 8^2 -4(1)(-6)=64+24=88$
88 is not a perfect square, so roots are irrational, meaning the polynomial cannot be factored into integer linear factors.

Answer:

The polynomial is prime.