Question

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


f(x)=x3−x2+6x−6




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


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

Answer 1

Answer: The correct option is 2.

Explanation:

The given polynomial is,

$f(x)=x^3-x^2+6x-6$

By rational root theorem 1 and -1 are possible roots of all polynomials.

put x=1

$f(x)=(1)^3-(1)^2+6(1)-6=0$

Since the value of f(x) is 0 at x=1. So 1 is a real root of the polynomial.

Divide the given polynomial by  (x-1) using synthetic division method.

$f(x)=(x-1)(x^2+6)$

Equate each factor equal to 0.

$x=1$

$x=\pm\sqrt{-6}$

So, the given polynomial have one real root and 2 non real roots. Since the function have one real root therefore the function intersects the x-axis at exactly one location.

Thus, the correct option is 2.

Answer 2

Answer:

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

Step-by-step explanation: