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

Question

Quizplus · Accepted Answer

To find the real zeros of the polynomial $f(x) = x^3 + 5x^2 - 4x - 20$, we can attempt to factor it or use synthetic division to test possible rational roots. The Rational Root Theorem suggests that possible rational zeros are factors of the constant term (-20) over factors of the leading coefficient (1), which are $\pm1, \pm2, \pm4, \pm5, \pm10, \pm20$. Testing these, we find that $x = -5$ is a root because $f(-5) = (-5)^3 + 5(-5)^2 - 4(-5) - 20 = 0$. Factoring or using synthetic division with $x + 5$ will reduce the polynomial to a quadratic, which can be solved to find the other roots. Factoring out $x + 5$ gives us a quadratic that can be solved to find the remaining roots, $x = 2$ and $x = -2$, each with multiplicity 1, confirming that all roots are real and distinct.

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

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