Intermediate Algebra Tutorial 30


Intermediate Algebra
Tutorial 30: Solve by Factoring

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deskLearning Objectives

 
After completing this tutorial, you should be able to:
  1. Write a polynomial equation in standard form.
  2. Use the zero factor property.
  3. Solve polynomial equations by factoring.




desk Introduction


In this tutorial we will be putting our factoring skills to the test.  We will be looking at solving polynomial equations, which include quadratic equations, by factoring.  Solving equations in general is a very essential part of Algebra.  So I guess we better get to it.

 

 

desk Tutorial

 
 

Polynomial Equation
 
A polynomial equation is one polynomial set equal to another polynomial.

The following is an example of a polynomial equation:

polynomial
 
  Standard Form of a Polynomial Equation
 

When a polynomial is set equal to 0, it is in standard form.

The following is an example of a polynomial equation in standard form:

polynomial
 
  Quadratic Equation in Standard Form

quartatic equation

where a does not equal 0.
 

The degree of a quadratic equation is 2.  This is a common type of equation. 


  Zero Factor Property

If ab = 0, then a = 0 or b = 0. 
 

0 is our magic number because the only way a product can become 0 is if at least one of its factors is 0. 

You can not guarantee what the factors would have to be if the product was set equal to any other number.  For example if ab = 1, then a = 5 and b = 1/5 or a = 3 and b = 1/3, etc.  But with the product set equal to 0, we can guarantee finding the solution by setting each factor equal to 0. 
 
  Solving a Polynomial Equation by Factoring
 

Step 1: Simplify if needed.

 
Clear any fractions or (   ) that are not in standard form in the problem.

 
Step 2: Write in standard form if needed .

 
If needed, use inverse operations to get the polynomial set equal to 0.

 
Step 3: Factor the Polynomial.

 
Use the factoring strategy shown in Tutorial 29: Factoring Special Products to factor the polynomial completely.

 
Step 4: Use the Zero Factor Property.

 
Set every factor equal to zero.

 
Step 5: Solve for the equations set up in step 4.

 


 
 

notebook Example 1:   Solve the equation example 1.

 
Step 1: Simplify if needed.

 
In this problem, there is no simplifying that we can do.  Even though we have ( ), the problem is already in standard form, so we do not need to multiply it out.

 
Step 2: Write in standard form if needed .

 
This problem is already written in standard form.

 
Step 3: Factor the Polynomial.

 
This problem is already factored.

 
Step 4: Use the Zero Factor Property

AND

Step 5: Solve for the equations set up in step 4.

 
example 1b
*Set 1st factor = 0
*Solve for x
 
 
 

*Set 2nd factor = 0
*Solve for x
 
 

*Set 3rd factor = 0
*Solve for x
 

The solutions to this equation are 0, 1, and -3.

Note that if we plug any one of these three numbers into the original equation, they would make the left side equal to the right side.
 
 
 

notebook Example 2:   Solve the equation example 2a.

 
Step 1: Simplify if needed.

 
In this problem, there is no simplifying that we can do.  Even though we have ( ), the problem is already in standard form, so we do not need to multiply it out.

 
Step 2: Write in standard form if needed.

 
This problem is already written in standard form.

 
Step 3: Factor the Polynomial.

 
This problem is already factored.

 
Step 4: Use the Zero Factor Property

AND

Step 5: Solve for the equations set up in step 4.

 
example 2b
*5 can never equal 0
 

*Set 2nd factor = 0
*Solve for x
 
 

*Set 3rd factor = 0
*Solve for x
 
 
 
 

 
 

Note that one of our factors is a constant 5.  A non-zero constant will never equal 0, so that factor does not give us a solution.

The solutions to this equation are - 4 and 5/2.

Note that if we plug either of these numbers into the original equation,  they would make the left side equal to the right side.
 
 
 

notebook Example 3:   Solve the equation example 3a.

 
Step 1: Simplify if needed.

 
In this problem, there is no simplifying that we can do. 

 
Step 2: Write in standard form if needed.

 
This problem is already written in standard form.

 
Step 3: Factor the Polynomial.

 
example 3b

*Factor the trinomial

 
Step 4: Use the Zero Factor Property

AND

Step 5: Solve for the equations set up in step 4.

 
example 3c
*Set 1st factor = 0
*Solve for x
 
 

*Set 2nd factor = 0
*Solve for x
 
 

The solutions to this equation are -3 and 1.

Note that if we plug either of these numbers into the original equation,  they would make the left side equal to the right side.
 
 
 

notebook Example 4:   Solve the equation example 4a.

 
Step 1: Simplify if needed.

 
We can simplify this problem by multiplying both sides of the equation by the LCD.  This will clear out the fractions and make it a nicer problem to work with.

 
example 4b
*Multiply both sides by LCD 10

 
Step 2: Write in standard form if needed.

 
This problem is not written in standard form.  We need to move the 4 over to the other side of this equation so we have the polynomial set equal to 0.

 
example 4c
*Get polynomial = 0 

 
Step 3: Factor the Polynomial.

 
example 4d
*Factor the difference of squares

 
Step 4: Use the Zero Factor Property

AND

Step 5: Solve for the equations set up in step 4.

 
example 4e
*Set 1st factor = 0
*Solve for y
 
 
 
 
 
 

*Set 2nd factor = 0
*Solve for y
 
 
 
 

 
 

The solutions to this equation are -2/3 and 2/3.

Note that if we plug either of these numbers into the original equation, they would make the left side equal to the right side.
 
 
 

notebook Example 5:   Solve the equation example 5a.

 
Step 1: Simplify if needed.

 
In this problem, there is no simplifying to do. 

 
 
Step 2: Write in standard form if needed.

 
This problem is not written in standard form.  We need to move the 2z over to the other side of this equation so we have the polynomial set equal to 0.

 
example 5b

*Get polynomial = 0 

 
Step 3: Factor the Polynomial.

 
example 5c

*Factor a GCF out
*Factor the trinomial

 
Step 4: Use the Zero Factor Property

AND

Step 5: Solve for the equations set up in step 4.

 
example 5d
*Set 1st factor = 0
 

*Set 2nd factor = 0
*Solve for z
 
 
 
 
 
 

*Set 3rd factor = 0
*Solve for z
 
 
 

 
 

The solutions to this equation are 1/4 and -2/3.

Note that if we plug any of these numbers into the original equation,  they would make the left side equal to the right side.
 
 

notebook Example 6:   Solve the equation example 6a.

 
Step 1: Simplify if needed.

 
On this problem, you need to be careful.  Since it is set equal to 24, it is NOT in standard form.  So we need to multiply the left side out and then get it into standard form.

 
example 6b

*Use the FOIL method to multiply

 
Step 2: Write in standard form if needed.

 
This problem is not written in standard form.  We need to move the 24 over to the other side of this equation so we have the polynomial set equal to 0.

 
example 6c

*Get polynomial = 0 

 
Step 3: Factor the Polynomial.

 
example 6d

*Factor the trinomial

 
Step 4: Use the Zero Factor Property

AND

Step 5: Solve for the equations set up in step 4.

 
example 6e
*Set 1st factor = 0
*Solve for t
 
 

*Set 2nd factor = 0
*Solve for t
 
 

The solutions to this equation are -7 and 4.

Note that if we plug either of these numbers into the original equation, they would make the left side equal to the right side.
 

 

 
desk Practice Problems

  
These are practice problems to help bring you to the next level.  It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it.  Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument.  In fact there is no such thing as too much practice.

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that  problem.  At the link you will find the answer as well as any steps that went into finding that answer.

 

pencil Practice Problems 1a - 1d: Solve each equation.

 

1a. problem 1a
(answer/discussion to 1a)
1b. problem 1b
(answer/discussion to 1b)

1c. problem 1c
(answer/discussion to 1c)

1d. problem 1d
(answer/discussion to 1d)

 

 

desk Need Extra Help on these Topics?


 
The following are webpages that can assist you in the topics that were covered on this page:
 

http://www.sosmath.com/algebra/quadraticeq/sobyfactor/sobyfactor.html 
This webpage helps with solving quadratic equations by factoring. 

http://www.mathpower.com/tut99.htm 
This webpage helps with solving quadratic equations by factoring. 

http://www.mathpower.com/tut105.htm 
This webpage helps with solving quadratic equations by factoring. 

http://www.mathpower.com/tut110.htm 
This webpage helps with solving quadratic equations by factoring. 
 

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.


 

WTAMU > Virtual Math Lab > Intermediate Algebra


Last revised on July 16, 2011 by Kim Seward.
All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.