Question

what are all the real and complex zeros of the polynomial function shown in the graph?​

Answer 1

Answer:

$x=3, -1-i, -1+i$

Step-by-step explanation:

We are given the graph of the function:

$f(x)=x^3-x^2-4x-6$

And we want to determine its real and complex roots.

First, notice that it crosses through the x-axis at 3. This means that (x-3) is a factor.

Hence, let's use synthetic division to factor. This yields:

3 |     1         -1        -4         -6            

  |_______(3)___(6)___(6)_________________

        1         2        2          0

Therefore,this yields:

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

The right-most term is not factorable. Thus, we will need to use the quadratic formula.

Zero Product Property:

$0=x-3\text{ or } 0=x^2+2x+2$

We will use the quadratic formula on the right. Our a=1, b=2, and c=2. Therefore:

$x=\frac{-(2)\pm\sqrt{(2)^2-4(1)(2)}}{2(1)}$

Evaluate:

$x=\frac{-2\pm\sqrt{-4}}{2}$

Simplify:

$x=\frac{-2\pm2i}{2}=-1\pm i$

Hence, our solutions are:

$x=3, -1-i, -1+i$