Find all real zeros of the polynomial $f ( x ) = x ^ { 4 } - 80 x ^ { 2 } + 1024$ and determine the mutiplicity of each.
A) $x = 64$, multiplicity $2 ; x = 16$, multiplicity 2
B) $x = 8$, multiplicity $2 ; x = 4$, multiplicity 2
C) $x = 64$, multiplicity $2 ; x = 4$, multiplicity 1
D) $x = - 8$, multiplicity $2 ; x = - 4$, multiplicity 2
E) $x = 8$, multiplicity $1 ; x = - 8$, multiplicity $1 ; x = 4$, multiplicity $1 ; x = - 4$, multiplicity 1

Question

Quizplus · Accepted Answer

The polynomial can be factored as $(x^2 - 64)^2 = (x - 8)^2(x + 8)^2$, leading to zeros at $x = 8$ and $x = -8$, each with multiplicity 2. However, there was an oversight in the explanation regarding the presence of $x = 4$ and $x = -4$ as zeros, which are not zeros of the given polynomial. The correct zeros are indeed $x = 8$ and $x = -8$, each with multiplicity 2, aligning with option B's description of the zeros but mistakenly identified under option E due to an error in the explanation process.

Find All Real Zeros of the Polynomial f(x)=x4−80x2+1024f ( x ) = x ^ { 4 } - 80 x ^ { 2 } + 1024f(x)=x4−80x2+1024

Find All Real Zeros of the Polynomial $f ( x ) = x ^ { 4 } - 80 x ^ { 2 } + 1024$