# Factoring

## Polynomial Equations

We have spent considerable time learning how to factor polynomials. We will now look at polynomial equations and solve them using factoring, if possible.
A polynomial equation is an equation that contains a polynomial expression. The degree of the polynomial equation is the degree of the polynomial.
We have already solved polynomial equations of degree one. Polynomial equations of degree one are linear equations are of the form 
We are now going to solve polynomial equations of degree two. A polynomial equation of degree two is called a quadratic equation. Listed below are some examples of quadratic equations:
The last equation doesn’t appear to have the variable squared, but when we simplify the expression on the left we will get 
The general form of a quadratic equation is  with  (If  then  and we are left with no quadratic term.)
To solve quadratic equations we need methods different from the ones we used in solving linear equations. We will look at one method here and then several others in a later chapter.

### Use the Zero Product Property

We will first solve some quadratic equations by using the Zero Product Property. The Zero Product Property says that if the product of two quantities is zero, then at least one of the quantities is zero. The only way to get a product equal to zero is to multiply by zero itself.
We will now use the Zero Product Property, to solve a quadratic equation.

### Solve Quadratic Equations by Factoring

The Zero Product Property works very nicely to solve quadratic equations. The quadratic equation must be factored, with zero isolated on one side. So we must be sure to start with the quadratic equation in standard form,  Then we must factor the expression on the left.
Before we factor, we must make sure the quadratic equation is in standard form.
Solving quadratic equations by factoring will make use of all the factoring techniques you have learned in this chapter! Do you recognize the special product pattern in the next example?
In the next example, the left side of the equation is factored, but the right side is not zero. In order to use the Zero Product Property, one side of the equation must be zero. We’ll multiply the factors and then write the equation in standard form.
In the next example, when we factor the quadratic equation we will get three factors. However the first factor is a constant. We know that factor cannot equal 0.
The Zero Product Property also applies to the product of three or more factors. If the product is zero, at least one of the factors must be zero. We can solve some equations of degree greater than two by using the Zero Product Property, just like we solved quadratic equations.

### Solve Equations with Polynomial Functions

As our study of polynomial functions continues, it will often be important to know when the function will have a certain value or what points lie on the graph of the function. Our work with the Zero Product Property will be help us find these answers.
The Zero Product Property also helps us determine where the function is zero. A value of x where the function is 0, is called a zero of the function.
When  the point  is a point on the graph. This point is an  of the graph. It is often important to know where the graph of a function crosses the axes. We will see some examples later.

### Solve Applications Modeled by Polynomial Equations

The problem-solving strategy we used earlier for applications that translate to linear equations will work just as well for applications that translate to polynomial equations. We will copy the problem-solving strategy here so we can use it for reference.
We will start with a number problem to get practice translating words into a polynomial equation.
Were you surprised by the pair of negative integers that is one of the solutions to the previous example? The product of the two positive integers and the product of the two negative integers both give positive results.
In some applications, negative solutions will result from the algebra, but will not be realistic for the situation.
In the next example, we will use the Pythagorean Theorem  This formula gives the relation between the legs and the hypotenuse of a right triangle.
We will use this formula to in the next example.
The next example uses the function that gives the height of an object as a function of time when it is thrown from 80 feet above the ground.

### Key Concepts

1. Polynomial Equation: A polynomial equation is an equation that contains a polynomial expression. The degree of the polynomial equation is the degree of the polynomial.
2. Quadratic Equation: An equation of the form  is called a quadratic equation.

3. Zero Product Property: If  then either  or  or both.
4. How to use the Zero Product Property


5. How to solve a quadratic equation by factoring.


6. Zero of a Function: For any function f, if  then x is a zero of the function.
7. How to use a problem solving strategy to solve word problems.



### Section Exercises


### Practice Makes Perfect

Use the Zero Product Property
In the following exercises, solve.
Solve Quadratic Equations by Factoring
In the following exercises, solve.
Solve Equations with Polynomial Functions
In the following exercises, solve.
In the following exercises, for each function, find: ⓐ the zeros of the function ⓑ the x-intercepts of the graph of the function ⓒ the y-intercept of the graph of the function.
Solve Applications Modeled by Quadratic Equations
In the following exercises, solve.

### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

### Chapter Review Exercises


### Greatest Common Factor and Factor by Grouping

Find the Greatest Common Factor of Two or More Expressions
In the following exercises, find the greatest common factor.
Factor the Greatest Common Factor from a Polynomial
In the following exercises, factor the greatest common factor from each polynomial.
Factor by Grouping
In the following exercises, factor by grouping.

### Factor Trinomials

Factor Trinomials of the Form 
In the following exercises, factor each trinomial of the form 
In the following examples, factor each trinomial of the form 
Factor Trinomials of the Form 
In the following exercises, factor completely using trial and error.
Factor Trinomials of the Form 
In the following exercises, factor.
Factor using substitution
In the following exercises, factor using substitution.

### Factor Special Products

Factor Perfect Square Trinomials
In the following exercises, factor completely using the perfect square trinomials pattern.
Factor Differences of Squares
In the following exercises, factor completely using the difference of squares pattern, if possible.
Factor Sums and Differences of Cubes
In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.

### General Strategy for Factoring Polynomials

Recognize and Use the Appropriate Method to Factor a Polynomial Completely
In the following exercises, factor completely.

### Polynomial Equations

Use the Zero Product Property
In the following exercises, solve.
Solve Quadratic Equations by Factoring
In the following exercises, solve.
Solve Equations with Polynomial Functions
In the following exercises, solve.
In each function, find: ⓐ the zeros of the function ⓑ the x-intercepts of the graph of the function ⓒ the y-intercept of the graph of the function.
Solve Applications Modeled by Quadratic Equations
In the following exercises, solve.

### Chapter Practice Test

In the following exercises, factor completely.
In the following exercises, solve