# Precalculus : Polynomial Functions

## Example Questions

### Example Question #1 : 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 #7 : 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 #8 : 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 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 )

(multiplicity of 2 on 0, multiplicity of 1 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 a solution as 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 #21 : Fundamental Theorem Of Algebra

Find a fourth-degree polynomial whose zeroes are , 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 #22 : 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 #1 : Find The Sum And Product Of The Zeros Of A Polynomial

Given , determine the sum and product of the zeros respectively.