Polynomial Division and Evaluation Problem

Vocabulary and concept flashcards detailing the step-by-step algebraic solution of a degree-3 polynomial division and evaluation problem.

Factored form of $P(x)$

$P(x) = a \times x(x - 1)(x + 1)$, derived from the common factors of the divisors $x^2 - x = x(x - 1)$ and $x^2 - 1 = (x - 1)(x + 1)$ for a degree $3$ polynomial.

Sum of the coefficients of $P(x + 1)$

The value evaluated by setting $x = 1$, giving $P(1 + 1) = P(2) = 18$.

Leading coefficient $a$ of $P(x)$

$a = 3$, determined from the equation $P(2) = a \times 2 \times 1 \times 3 = 6a = 18$.

Constant term of $P(6x - 4)$

The value evaluated by setting $x = 0$, which yields $P(6(0) - 4) = P(-4)$.

Final value of $P(-4)$

$-180$, obtained by computing $3 \times (-4) \times (-5) \times (-3)$, corresponding to option C.