Find all real zeros of the polynomial $f ( x ) = x ^ { 4 } + 12 x ^ { 3 } + 27 x ^ { 2 }$ and determine the multiplicity of each.
A) $x = 0 \text {, multiplicity } 1 ; x = 9 , \text { multiplicity } 1 ; x = - 9 , \text { multiplicity } 1 ; x = 3 \text {, multiplicity } 1$
B) $x = 0 \text {, multiplicity } 2 ; x = 9 \text {, multiplicity } 1 ; x = 3 \text {, multiplicity } 1$
C) $x = 0 \text {, multiplicity } 2 ; x = - 9 \text {, multiplicity } 1 ; x = - 3 \text {, multiplicity } 1$
D) $x = 9 , \text { multiplicity } 2 ; x = 3 , \text { multiplicity } 2$

Question

Quizplus · Accepted Answer

Factoring the polynomial $f(x) = x^4 + 12x^3 + 27x^2$, we get $f(x) = x^2(x^2 + 12x + 27)$. Further factoring gives $f(x) = x^2(x + 3)(x + 9)$, indicating zeros at $x = 0$ with multiplicity 2, $x = -3$, and $x = -9$ each with multiplicity 1.

Find All Real Zeros of the Polynomial f(x)=x4+12x3+27x2f ( x ) = x ^ { 4 } + 12 x ^ { 3 } + 27 x ^ { 2 }f(x)=x4+12x3+27x2

Find All Real Zeros of the Polynomial $f ( x ) = x ^ { 4 } + 12 x ^ { 3 } + 27 x ^ { 2 }$