Find all real zeros of the polynomial $f ( x ) = x ^ { 3 } + 3 x ^ { 2 } - 49 x - 147$ and determine the mutiplicity of each.
A) $x = 7$, multiplicity $2 ; x = - 3$, multiplicity 1
B) $x = 7$, multiplicity $1 ; x = - 7$, multiplicity $1 ; x = - 3$, multiplicity 1
C) $x = - 3$, multiplicity $2 ; x = - 7$, multiplicity 1
D) $x = - 7$, multiplicity $1 ; x = 3$, multiplicity $1 ; x = - 3$, multiplicity 1
E) $\quad x = - 3$, multiplicity 3

Question

Quizplus · Accepted Answer

We can use synthetic division or guess and check to find that $(x-7)$ and $(x+7)$ are factors of the polynomial, while $-3$ is a root of multiplicity 1. Therefore, the real zeros are $x=7$ and $x=-7$ with multiplicity 1 each, and $x=-3$ with multiplicity 1. This matches option B.

Find All Real Zeros of the Polynomial f(x)=x3+3x2−49x−147f ( x ) = x ^ { 3 } + 3 x ^ { 2 } - 49 x - 147f(x)=x3+3x2−49x−147

Find All Real Zeros of the Polynomial $f ( x ) = x ^ { 3 } + 3 x ^ { 2 } - 49 x - 147$