Find all the real zeros of the polynomial function $f ( x ) = x ^ { 3 } + 5 x ^ { 2 } - 4 x - 20$ and determine the multiplicity of each.
A) $x = 2$ , multiplicity 2; $x = - 5$ , multiplicity 1
B) $x = 2$ , multiplicity 1; $x = - 2$ , multiplicity 1; $x = - 5$ , multiplicity 1
C) $x = - 5$ , multiplicity 2; $x = - 2$ , multiplicity 1
D) $x = - 2$ , multiplicity 1; $x = 5$ , multiplicity 1; $x = - 5$ , multiplicity 1
E) $x = - 5$ , multiplicity 3

Question

Quizplus · Accepted Answer

To find the real zeros of the polynomial $f(x) = x^3 + 5x^2 - 4x - 20$, we can use synthetic division or factoring if possible. Factoring by grouping or trying rational roots can be helpful. One can notice that $x = -5$ is a root because $f(-5) = (-5)^3 + 5(-5)^2 - 4(-5) - 20 = 0$. After dividing the polynomial by $x + 5$, we get $x^2 - 4$, which can be further factored into $(x - 2)(x + 2)$. Thus, the real zeros are $x = -5$, $x = 2$, and $x = -2$, each with multiplicity 1, matching option B.

Find All the Real Zeros of the Polynomial Function f(x)=x3+5x2−4x−20f ( x ) = x ^ { 3 } + 5 x ^ { 2 } - 4 x - 20f(x)=x3+5x2−4x−20

Find All the Real Zeros of the Polynomial Function $f ( x ) = x ^ { 3 } + 5 x ^ { 2 } - 4 x - 20$