Question

20 POINTS

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

f(x) = x^3 + x^2 − 8x − 8


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

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

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

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

Answer 1

answer : option C

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

Lets find the number of zeros by factoring

`0 = x^3 + x^2 - 8x - 8`

Group first two terms and last two terms

`0 = (x^3 + x^2) (- 8x - 8)`

Factor out GCF from each group

`0 = x^2(x + 1)-8(x + 1)`

`0 = (x^2-8)(x + 1)`

Now we set each factor =0 and solve for x

0 = (x^2-8) and (x + 1)=0

`x^2 = 8` and x = -1

`x= 2+-√(2)` and x= -1

So we have three real zeros

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