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

Step-by-step explanation:

We are given the graph of the function:

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:

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

Zero Product Property:

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

Evaluate:

Simplify:

Hence, our solutions are:

x =±sqrt(1/3)

step-by-step explanation:

x+4/6x=1/x

multiply each side by 6x to get rid of the fractions

6x*(x+4/6x)=1/x*6x

distribute

6x^2 +4 = 6

subtract 4 from each side

6x^2 +4-4 = 6-4

6x^2 =2

divide by 6

6x^2/6 = 2/6

x^2 = 1/3

take the square root of each side

sqrt(x^2) = ±sqrt(1/3)

x =±sqrt(1/3)

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step-by-step explanation: