Find all real zeros of the polynomial $f ( x ) = x ^ { 4 } + 8 x ^ { 3 } + 12 x ^ { 2 }$ and determine the mutiplicity of each.
A) $x = 0$, multiplicity $2 ; x = - 2$, multiplicity $1 ; x = - 6$, multiplicity 1
B) $x = 2$, multiplicity $2 ; x = 6$, multiplicity 2
C) $x = 0$, multiplicity $2 ; x = 2$, multiplicity $1 ; x = 6$, multiplicity 1
D) $x = - 2$, multiplicity $2 ; x = - 6$, multiplicity 2
E) $x = 0$, multiplicity $1 ; x = 2$, multiplicity $1 ; x = - 2$, multiplicity $1 ; x = 6$, multiplicity 1

Question

Quizplus · Accepted Answer

Factoring the polynomial $f(x) = x^4 + 8x^3 + 12x^2$ gives $x^2(x^2 + 8x + 12)$. Further factoring $x^2 + 8x + 12$ gives $x^2(x + 6)(x + 2)$. Thus, the real zeros are $x = 0$ with multiplicity 2, $x = -2$ with multiplicity 1, and $x = -6$ with multiplicity 1.

Find All Real Zeros of the Polynomial f(x)=x4+8x3+12x2f ( x ) = x ^ { 4 } + 8 x ^ { 3 } + 12 x ^ { 2 }f(x)=x4+8x3+12x2

Find All Real Zeros of the Polynomial $f ( x ) = x ^ { 4 } + 8 x ^ { 3 } + 12 x ^ { 2 }$