Simplifying and Factoring Algebra equations

SIMPLIFYING AND FACTORING ALGEBRA EQUATIONS

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Intermediate Algebra
Tutorial 30:
Solving By Factoring

Learning Objectives

After completing this tutorial, you should be able to:

Write a polynomial equation in standard form.
Use the zero factor property.
Solve polynomial equations by factoring.

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. After completing this tutorial, you will be a master at solving polynomial equations, unlike the ones in this Close to Home comic. Solving equations in general is a very essential part of Algebra. So I guess we better get to it.

Tutorial

Polynomial Equation

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

The following is an example of a polynomial equation:

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:

Quadratic Equation in Standard Form

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.

Example 1: Solve the equation

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.

*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.

Example 2: Solve the equation

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.

*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.

Example 3: Solve the equation

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.

*Factor the trinomial

Step 4: Use the Zero Factor Property

AND

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

*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.

Example 4: Solve the equation

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.

*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.

*Get polynomial = 0

Step 3: Factor the Polynomial.

*Factor the difference of squares

Step 4: Use the Zero Factor Property

AND

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

*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.

Example 5: Solve the equation

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.

*Get polynomial = 0

Step 3: Factor the Polynomial.

*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.

*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.

Example 6: Solve the equation

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.

*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.

*Get polynomial = 0

Step 3: Factor the Polynomial.

*Factor the trinomial

Step 4: Use the Zero Factor Property

AND

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

*Set 1st factor = 0
*Solve fort

*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.

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.