Use synthetic division and the Factor Theorem to determine whether or not $4 x + 3$ is a factor of $P ( x ) = - 12 x ^ { 3 } - 40 x ^ { 2 } + 52 x + 80$ .
A)No, $4 x + 3$ is not a factor.
B)Yes, $4 x + 3$ is a factor.

Question

Quizplus · Accepted Answer

To determine if $4x + 3$ is a factor of $P(x) = -12x^3 - 40x^2 + 52x + 80$, we use the Factor Theorem. This involves evaluating $P(x)$ at the root of $4x + 3 = 0$, which is $x = -\frac{3}{4}$. If $P(-\frac{3}{4}) = 0$, then $4x + 3$ is a factor. Using synthetic division or direct substitution into $P(x)$, we find that $P(-\frac{3}{4})$ does not equal 0, indicating that $4x + 3$ is not a factor of $P(x)$.

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