Find the real zeros of the polynomial function $f ( x ) = x ^ { 5 } - 16 x ^ { 4 } + 64 x ^ { 3 }$ and determine the multiplicity of each.
A) $x = 0$ , multiplicity 2; $x = 8$ , multiplicity 3
B) $x = 0$ , multiplicity 2; $x = 8$ , multiplicity 1; $x = - 8$ , multiplicity 2
C) $x = 0$ , multiplicity 3; $x = 8$ , multiplicity 2
D) $x = 0$ , multiplicity 3; $x = - 8$ , multiplicity 2
E) $x = 0$ , multiplicity 2; $x = 8$ , multiplicity 2; $x = - 8$ , multiplicity 1

Question

Quizplus · Accepted Answer

Factoring the given polynomial $f(x) = x^5 - 16x^4 + 64x^3$ as $x^3(x^2 - 16x + 64)$, we see $x = 0$ is a root with multiplicity 3 (since $x^3$ term). The quadratic can be factored as $(x-8)^2$, giving $x = 8$ as a root with multiplicity 2.

Find the Real Zeros of the Polynomial Function f(x)=x5−16x4+64x3f ( x ) = x ^ { 5 } - 16 x ^ { 4 } + 64 x ^ { 3 }f(x)=x5−16x4+64x3

Find the Real Zeros of the Polynomial Function $f ( x ) = x ^ { 5 } - 16 x ^ { 4 } + 64 x ^ { 3 }$