# 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 point . Find the other two solutions.

Explanation:

Since we know that one of the zeros of this polynomial is 3, we know that one of the factors is . To find the other two zeros, we can divide the original polynomial by , either with long division or with synthetic division:

This gives us the second factor of . We can get our solutions by using the quadratic formula:

### Example Question #5 : Find Complex Zeros Of A Polynomial Using The Fundamental Theorem Of Algebra

Find all the real and complex zeroes of the following equation:

Explanation:

First, factorize the equation using grouping of common terms:

Next, setting each expression in parenthesis equal to zero yields the answers.

### Example Question #6 : Find Complex Zeros Of A Polynomial Using The Fundamental Theorem Of Algebra

Find all the zeroes of the following equation and their multiplicity:

(multiplicity of 2 on 0, multiplicity of 1 on

(multiplicity of 1 on 0, multiplicity of 2 on

(multiplicity of 2 on 0, multiplicity of 1 on

(multiplicity of 1 on 0, multiplicity of 2 on

(multiplicity of 1 on 0, multiplicity of 2 on

Explanation:

First, pull out the common t and then factorize using quadratic factoring rules:

This equation has solutions at two values: when  and when

Therefore, But since the degree on the former equation is one and the degree on the latter equation is two, the multiplicities are 1 and 2 respectively.

### Example Question #6 : Find Complex Zeros Of A Polynomial Using The Fundamental Theorem Of Algebra

Find a fourth degree polynomial whose zeroes are -2, 5, and

Explanation:

This one is a bit of a journey. The expressions for the first two zeroes are easily calculated,  and  respectively. The last expression must be broken up into two equations:

which are then set equal to zero to yield the expressions  and

Finally, we multiply together all of the parenthesized expressions, which multiplies out to

### Example Question #7 : Find Complex Zeros Of A Polynomial Using The Fundamental Theorem Of Algebra

The third degree polynomial expression  has a real zero at Find all of the complex zeroes.

Explanation:

First, factor the expression by grouping:

To find the complex zeroes, set the term  equal to zero:

### Example Question #9 : Find Complex Zeros Of A Polynomial Using The Fundamental Theorem Of Algebra

Find all the real and complex zeros of the following equation:

Explanation:

First, factorize the equation using grouping of common terms:

Next, setting each expression in parentheses equal to zero yields the answers.

### Example Question #10 : Find Complex Zeros Of A Polynomial Using The Fundamental Theorem Of Algebra

Find all the zeroes of the following equation and their multiplicity:

(multiplicity of 1 on 0, multiplicity of 2 on )

(multiplicity of 2 on 0, multiplicity of 1 on )

(multiplicity of 1 on 0, multiplicity of 2 on )

(multiplicity of 2 on 0, multiplicity of 1 on )