# Precalculus : Find Complex Zeros of a Polynomial Using the Fundamental Theorem of Algebra

## Example Questions

### Example Question #1 : Find Complex Zeros Of A Polynomial Using The Fundamental Theorem Of Algebra

What are the roots of

including complex roots, if they exist?

Explanation:

One of the roots is  because if we plug in 1, we get 0. We can factor the polynomial as

So now we solve the roots of .

The root will not be real.

The roots of this polynomial are .

So, the roots are

### Example Question #2 : Find Complex Zeros Of A Polynomial Using The Fundamental Theorem Of Algebra

The polynomial has a real zero at 1.5. Find the other two zeros.

Explanation:

If this polynomial has a real zero at 1.5, that means that the polynomial has a factor that when set equal to zero has a solution of . We can figure out what this is this way:

multiply both sides by 2

is the factor

Now that we have one factor, we can divide to find the other two solutions:

To finish solving, we can use the quadratic formula with the resulting quadratic,

### Example Question #3 : Find Complex Zeros Of A Polynomial Using The Fundamental Theorem Of Algebra

If the real zero of the polynomial is 3, what are the complex zeros?

Explanation:

We know that the real zero of this polynomial is 3, so one of the factors must be . To find the other factors, we can divide the original polynomial by , either by long division or synthetic division:

This gives us a second factor of which we can solve using the quadratic formula:

### Example Question #4 : Find Complex Zeros Of A Polynomial Using The Fundamental Theorem Of Algebra

The polynomial intersects the x-axis at 3. Find the other two solutions.