FINDING ZEROS OF POLYNOMIAL FUNCTIONS USING FACTORING

Copyright by Ingrid Stewart, Ph.D.  Please Send Questions and Comments to ingrid.stewart@csn.edu.

Zeros (see definition in previous lecture) are often found so that we can better analyze the graphs of polynomial functions.  The Zeros of quadratic functions and certain polynomial functions of degree higher than 2, can be found using factoring, the Square Root Property, or the Quadratic Formula

Finding Zeros of Polynomial Functions using Factoring (You learned how to do this in Intermediate Algebra!!!)

A.   Factoring out the Greatest Common Factor

  • Set the function equal to 0.
  • Find the greatest common numeric factor of all of the coefficients.
  • Find the greatest common factor of all of the variables.
  • Divide out the greatest common factor.
  • Set each factor equal to 0 and solve to find the Zeros.

B.   Factoring relative to the integers using the Grouping Method.

  • Set the function equal to 0.
  • Collect the terms in an expression into two groups.
  • Factor out the greatest common factor from each group.  You should end up with an identical multinomial factor in each term.  If not, regroup, then factor out the greatest common factor again.  If you still don't end up with an identical multinomial factor in each term, the expression is NOT factorable by grouping.
  • Factor out the identical multinomial factor from each term.
  • Set each factor equal to 0 and solve to find the Zeros.

C.   Factoring the trinomial relative to the integers.  Assume a = 1.

  • Set the function equal to 0.
  • Using "educated guessing", find two integers whose product equals c and whose sum equals b.  These integers are always factors of c.
  • Form a product of factors containing the variable and each of the integers found.
  • Set each factor equal to 0 and solve to find the Zeros.

D.   Factoring the Trinomial relative to the Integers.  Assume a 1.

  • Set the function equal to 0.
  • Find two integers whose product equals ac and whose sum equals b.  These integers are always factors of ac.
  • Replace the coefficient b of the middle term bx in with the sum of the integers and distribute the variable.
  • Use the factoring by grouping method to find a product of two factors.
  • Set each factor equal to 0 and solve to find the Zeros.

E.   Factoring "special" polynomials relative to the Integers.

  • Set the function equal to 0.
  • Use the following formulas to factor the special polynomials:

Difference of Squares:    

Difference of Cubes:      

Sum of Cubes:               

  • Set each factor equal to 0 and solve to find the Zeros.

Please note that a Sums of Squares cannot be factored relative to the integers.  The factors of Sums of Squares are imaginary!  This will be discussed at a later time!

F.   Factoring polynomials that are "quadratic" in form relative to the integers.

Examples: OR

Follow the instructions in C) and D) above.

Example 1:

Find the Zeros of the polynomial function.

The degree of the polynomial is 3, therefore, we must find 3 Zeros, not necessarily distinct. 

  

Here, we will factor out the greatest common factor to get

Then, by the Zero Product Principle we know that or  or .

Therefore, , , and 

The Zeros are 5, 3, and 0Following is the graph of the function.  Since 5, 3, and 0 are real numbers, they are also the x-coordinate of x-intercepts of the graph of the function.

Below is the graph of the polynomial function:

Example 2: 

Let's find the Zeros of the polynomial function.

The degree of the polynomial is 4, therefore, we must find 4 Zeros, not necessarily distinct. 

.

Here, we will factor out the greatest common factor to get

Then, by the Zero Product Principle we get two quadratic equations or .

They can best be solved by using the Square Root Property.

(note that there is no + 0 or 0) or , which results in .

The Zeros are 0 (multiplicity 2) !!!, 3i, and 3iFollowing is the graph of the function.  Since 0 is a real number, it is also the x-coordinate of x-intercept of the graph of the function.  The two imaginary Zeros are not related to x-intercepts, however, they still help influence the shape of the graph.

Also notice, that due to the even multiplicity of 2 at (0, 0), the graph of the function touches the x-axis at (0, 0).  

Below is the graph of the polynomial function:

Example 3: 

Let's find the Zeros of the polynomial function.

Since the degree of the polynomial is four, we must find four Zeros, not necessarily distinct. 

Factoring might be accomplished easier, if we first multiply both sides of the equation by 1.

Next, we will factor out the greatest common factor to get

Then, by the Zero Product Principle we get two quadratic equations or .

They can best be solved by using the Square Root Property.

(note that there is no + 0 or 0) or , which results in .

The Zeros are 0 (multiplicity 2) !!!, 4, and 4 Following is the graph of the function.  Since 0, 4, and 4 are real numbers, they are also the x-coordinate of x-intercepts of the graph of the function.

Also notice, that due to the even multiplicity of 2 at (0, 0), the graph of the function touches the x-axis at (0, 0).  

Below is the graph of the polynomial function:

Example 4: 

Find the Zeros of the polynomial function .

The degree of the polynomial is 3, therefore, we must find 3 Zeros, not necessarily distinct. 

 

In this case, factoring is not readily apparent.  Therefore, we will try to group two terms and then we will try to factor the greatest common factor out of each group to see if this will allow us to achieve a common factor! 

Arbitrarily, let's make the following two groups.  Note that we connected them with a plus sign!

Now, we'll factor the greatest common factor out of each group.

We can see that each group has the common factor (2x + 3), which we will now factor out as follows:

Notice that the second factor is a Difference of Squares, which can be further factored to get

Using the Zero Product Principle, we get or or .

Therefore, , , and 

The Zeros are 2, , and 2.  Following is the graph of the function.  Since 2, , and 2 are real numbers, they are also the x-coordinate of x-intercepts of the graph of the function.

Below is the graph of the polynomial function:

Example 5: 

Find the Zeros of the polynomial function .

The degree of the polynomial is 4, therefore, we must find 4 Zeros, not necessarily distinct. 

This equation is said to be "like" the quadratic equation with .

We can factor the equation as follows:

Then, by the Zero Product Principle we get two quadratic equations or .

They can best be solved by using the Square Root Property.

or .

and or

The Zeros are 3, 3, i, and i

Following is the graph of the function.  Since 3 and 3 are real numbers, they are also the x-coordinate of x-intercept of the graph of the function.  The two imaginary Zeros are not related to x-intercepts, however, they still help influence the shape of the graph.

Below is the graph of the polynomial function:

Example 6: 

Let's find the Zeros of the polynomial function .

The degree of the polynomial is 3, therefore, we must find 3 Zeros, not necessarily distinct. 

Notice that we are dealing with a Difference of Cubes!

In our case, a = 3, therefore, we can factor as follows:

By utilizing the Zero Property Principle, we get

x - 3 = 0 or

Solving the first equation, we get x = 3.
To solve the second equation we must use the Quadratic Formula with
a = 1, b = 3, and c = 9.

Please note that it is customary and standard to place the imaginary number i in front of radicals instead of after them as is done with rational numbers.

The Zeros are 3, , and

The real Zero is the x-coordinate of the x-intercept of the graph of the function.  The two imaginary Zeros are not related to x-intercepts, however, they still help influence the shape of the graph.

Below is the graph of the polynomial function: