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Factoring 101

Factoring Pep-Talk

Factoring is an art.  There is no right or wrong way to think about factoring, there is no formula to apply, there is no single method that works… Many methods work.  Many lines of thinking work.  Factoring is hard.  But, I will help you through it.

There are many ways to factor, many things to factor, many uses of factoring, many, many, many… I am going to start us at square 1.

Why factor?


The main point of factoring is to get an expression to look like (something1)*(something2) = 0.  If two things multiply to equal 0, we know that one of those things must be 0.  Right?  If a*b = 0, either a or b must be 0… You can’t have 5*4 = 0….

Let’s factor:

When you see an equation that involves x², think “factor”

Example 1:

x² + 7x + 12 = 0

Step 1: Set the equation equal to zero.

Our equation already equals zero.. Step 1 complete.

Step 2:  Set up your parentheses

(x       )(x      ) = 0

We know there must be an “x” and an “x” in the first spot of each factor…. Just so you know: (       ) is considered to be a “factor” in my language.  As I was saying, we know there must be an “x” and an “x” because the first two positions must equal x² when multiplied together.

Step 3:  Figure out the signs in your factors

(x       )(x      ) = 0

We are either using a + & -, + & + or – & -.  How do we know which to use?

Look at the original problem:  + 7x + 12 = 0

Our two signs must multiply to give a + and combine (either through addition or subtraction) to get a +.  Our only options that multiply to give a positive and also combine to give a positive are + & +.  Therefore, we have:

(x +   )(x +   ) = 0

Step 4:  Figure out the numbers that go in the factors

Look at the original problem: + 7x + 12 = 0

Our two numbers must always multiply to give the back number (+12) and combine to give the middle number (+7).  This is where our “guessing game” begins.  What two numbers multiply to give 12 and combine to give 7?  +3 & +4!  So:

(x + 3)(x + 4) = 0

Tada!  The factoring part is done.  Now we just solve the problem:

(x + 3)(x + 4) = 0…. So:

(x + 3) = 0    OR (x + 4) = 0

x = -3    OR x = -4.

PHEW!

Example 2:

x² – 3x = 10

Step 1: Set the equation equal to zero.

Subtract 10 from each side to get:

x² – 3x – 10 = 0

Step 2:  Set up your parentheses

(x       )(x      ) = 0

Step 3:  Figure out the signs in your factors

(x       )(x      ) = 0

Options: + & -, + & + or – & -.

Look at the problem:  3x 10 = 0

Our two signs must multiply to give a and combine to get a .  Our only options that multiply to give a negative are + & -.  Therefore, we have:

(x +   )(x –   ) = 0

Step 4:  Figure out the numbers that go in the factors

Look at the original problem: 3x – 10 = 0

Our two numbers must always multiply to give the back number (-10) and combine to give the middle number (-3).  Guessing time!  What two numbers multiply to give -10 and combine to give -3?  -5 & +2!  So:

(x – 5)(x + 2) = 0

Tada!  The factoring part is done.  Now we just solve the problem:

(x – 5)(x + 2) = 0…. So:

(x – 5) = 0    OR (x + 2) = 0

x = 5    OR x = -2.

This concludes Factoring 101.  Stay tuned for Factoring 201 where we talking about different factoring methods and how to factor where there is a “number” in front of the x² term (i.e. 2x² – 2x – 4).  This is just to get you started!

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