Question

Which answer best describes the complex zeros of the polynomial function?

f(x)=x3+x2−8x−8




The function has two real zeros and one nonreal zero. The graph of the function intersects the x-axis at exactly one location.


The function has one real zero and two nonreal zeros. The graph of the function intersects the x-axis at exactly one location.

The function has three real zeros. The graph of the function intersects the x-axis at exactly three locations.


The function has one real zero and two nonreal zeros. The graph of the function intersects the x-axis at exactly two locations.

Answer 1

The answer best describes the zeros of the polynomial function f(x)=x^3+x^2−8x−8 is third option:

The function has three real zeros. The graph of the function intersects the x-axis at exactly three locations.

Explanation:

Using Ruffini:

f(x)=x^3+x^2-8x-8

! 1 1 -8 -8

-1 !___ -1___0__8

1 0 -8 0

x^2 -8

f(x)=(x+1)(x^2-8)

Factoring the quadratic term x^2-8, using:

a^2-b^2=(a+b)(a-b)

with:

a^2=x^2→sqrt(a^2)=sqrt(x^2)→a=x

b^2=8→sqrt(b^2)=sqrt(8)→b=sqrt(4*2)→b=sqrt(4) sqrt(2)→b=2 sqrt(2)

x^2-8=(x+2 sqrt(2))(x-2 sqrt(2))

f(x)=(x+1)(x+2 sqrt(2))(x-2 sqrt(2))

Zeros of the polynomial are: x=-1, x=-2 sqrt(2), and x=2 sqrt(2)

The function has three real zeros. The graph of the function intersects the x-axis at exactly three locations.