Find all real zeros of the polynomial $f ( x ) = x ^ { 4 } + 7 x ^ { 3 } + 6 x ^ { 2 }$ and determine the multiplicity of each.
A) $x = 0 \text {, multiplicity } 2 ; x = - 1 \text {, multiplicity } 1 ; x = - 6 \text {, multiplicity } 1$
B) $x = 1 , \text { multiplicity } 2 ; x = 6 , \text { multiplicity } 2$
C) $x = 0 \text {, multiplicity } 2 ; x = 1 \text {, multiplicity } 1 ; x = 6 \text {, multiplicity } 1$
D) $x = - 1 , \text { multiplicity } 2 ; x = - 6 , \text { multiplicity } 2$
E) $x = 0 , \text { multiplicity } 1 ; x = 1 , \text { multiplicity } 1 ; x = - 1 , \text { multiplicity } 1 ; x = 6 \text {, multiplicity } 1$

Question

Quizplus · Accepted Answer

Factoring $f(x) = x^4 + 7x^3 + 6x^2$, we get $f(x) = x^2(x^2 + 7x + 6)$. Further factoring gives $f(x) = x^2(x + 6)(x + 1)$, indicating zeros at $x = 0$ with multiplicity 2, $x = -6$ and $x = -1$ each with multiplicity 1.

Find All Real Zeros of the Polynomial f(x)=x4+7x3+6x2f ( x ) = x ^ { 4 } + 7 x ^ { 3 } + 6 x ^ { 2 }f(x)=x4+7x3+6x2

Find All Real Zeros of the Polynomial $f ( x ) = x ^ { 4 } + 7 x ^ { 3 } + 6 x ^ { 2 }$