Factor f(x) into linear factors given that k is a zero of f(x).
-$f ( x ) = x ^ { 3 } - 48 x - 128 ; k = - 4$ (multiplicity 2 $)$
A) $f ( x ) = ( x - 4 ) ^ { 2 } ( x - 8 )$
B) $f ( x ) = ( x + 4 ) ^ { 2 } ( x + 8 )$
C) $f ( x ) = ( x + 4 ) ^ { 2 } ( x - 8 )$
D) $f ( x ) = ( x + 8 ) ( x + 4 ) ( x - 4 )$

Question

Quizplus · Accepted Answer

Since $k = -4$ is a zero with multiplicity 2, $f(x)$ must have $(x + 4)^2$ as a factor. The remaining linear factor can be found by polynomial division or by using the fact that the sum of the roots equals the coefficient of $x^2$ (which is 0 here, indicating the remaining root must be $8$ to balance the $-4$ (twice) for a sum of 0). Thus, the correct factorization is $f(x) = (x + 4)^2(x - 8)$.

Factor F(x) into Linear Factors Given That K Is a Zero