Find all real zeros of the polynomial $f ( x ) = x ^ { 4 } + 13 x ^ { 3 } + 40 x ^ { 2 }$ and determine the mutiplicity of each.
A) $x = 0$, multiplicity $2 ; x = - 8$, multiplicity $1 ; x = - 5$, multiplicity 1
B) $\quad x = 8$, multiplicity $2 ; x = 5$, multiplicity 2
C) $x = 0$, multiplicity $2 ; x = 8$, multiplicity $1 ; x = 5$, multiplicity 1
D) $x = - 8$, multiplicity $2 ; x = - 5$, multiplicity 2
E) $\quad x = 0$, multiplicity $1 ; x = 8$, multiplicity $1 ; x = - 8$, multiplicity $1 ; x = 5$, multiplicity 1

Question

Quizplus · Accepted Answer

First, factor out the greatest common factor, which is $x^2$, to get $x^2(x^2 + 13x + 40)$. The quadratic can be factored further into $x^2(x + 8)(x + 5)$. This gives zeros at $x = 0$ with multiplicity 2, $x = -8$ and $x = -5$ each with multiplicity 1.

Find All Real Zeros of the Polynomial f(x)=x4+13x3+40x2f ( x ) = x ^ { 4 } + 13 x ^ { 3 } + 40 x ^ { 2 }f(x)=x4+13x3+40x2

Find All Real Zeros of the Polynomial $f ( x ) = x ^ { 4 } + 13 x ^ { 3 } + 40 x ^ { 2 }$