Factoring
   

   

 Math Help -> Procedures -> Factoring 

This section has some special hints and tips for 6th, 7th, and 8th grade math students.  One of the things that befuddles many 8th grade algebra students is "factoring", which I present on this page.  If this page is too hard for you, try "Very Basic Factoring" first, then come back here.

Contents

Very Basic Factoring
Pairs of Factors
Factoring Trinomials
Factoring: The Quadratic Formula
Cubic Formula
Factoring: Misc Topics

Jump directly to one of these topics, or else
read each page in sequence -- there's a link to
the next page at the bottom of each page.

Factoring a number means finding two other numbers (other than one and the number itself) whose product is the original number.

Factoring an expression means finding two other expressions (other than one and the expression itself) whose product is the original expression.  An expression is made up of symbols and numbers that are added, subtracted, multiplied and divided by one another.

Time for an example...

EXAMPLE 1:

Consider x2 + 2xy + y2

Can it be factored?  In other words, can there be two expressions that, when multiplied together, result in this expression?  The answer is yes.  Click here for the answer.

The expression in Example 1 not could not only be factored, but the two factors were the same -- in other words, example 1 was a "perfect square trinomial".  What does that mean, exactly?  In other words, how can one recognize a perfect square trinomial?  Here are the clues:

How to recognize a PERFECT SQUARE TRINOMIAL
  1. It has three terms
  2. Two of the terms are, themselves, perfect squares, and have the same sign
  3. The third term is twice the product of the square roots of the other two terms

That third requirement is quite a mouthful, I know.  But when you're checking the second requirement, keep the square roots in mind.  Multiply them together, and double it.  If that's the third term -- usually called the "middle" term -- then this is a perfect square trinomial.  (Remember: Every perfect square has two square roots.  I mean 16 is the square of 4 and also the square of -4.  So the middle term might be positive or negative -- it's a perfect square trinomial in either case.)

EXAMPLE 2:

Consider 4x2 - 12x + 9

Do you see this is a perfect square trinomial?  What is the square root of 4x2 - 12x + 9?.  Click here for the answer.

Now let's look at what happens when we multiply two expressions of the form

(a + b) (a - b)

Using the F.O.I.L. method, we multiply together the first terms, then the outer, inner, and last terms, and add 'em up:

a2 + ab - ab - b2

Do you see that the middle terms cancel each other out?

a2 - b2

By now you might have guessed how to factor the difference of two squares.  Let's do an example:

 Try this example:

EXAMPLE 3:

Consider 4x2 - 36

Do you see this is the difference of two squares?  Click here for the answer.

 
How to recognize THE DIFFERENCE OF TWO SQUARES
  1. There are two terms
  2. Both terms are perfect squares, and have opposite signs

Now we can factor a perfect square trinomial, and we can factor the difference of two squares.  What if one of those two squares is itself a perfect square trinomial?  Pause a minute to let that question sink in.

OK, now an example that combines the difference of two squares with a perfect square trinomial.

EXAMPLE 4:

Consider 4x2 + 12x + 9 - 49w2

Do you see this is the difference of two squares?  One of the squares is a perfect square trinomial.  Click here for the answer.

Now, when you need to factor an expression with four terms, look at them closely to see if it's the difference of a perfect square trinomial and another perfect square.

How to recognize the combination of A PERFECT SQUARE TRINOMIAL
and THE DIFFERENCE OF TWO SQUARES
  1. There are four terms
  2. Three of the terms are perfect squares, and one has the opposite sign from the others.
  3. Three of terms form a perfect square trinomial (these are the two perfect squares with the same sign and the one term that isn't a perfect square).

It's important that you read and understand how we factor expressions like this, because we will make use of the perfect square trinomial and the difference of squares when we derive the famous quadratic formula.

If you read this page, but you don't really understand it yet, try "Very Basic Factoring" then come back to this page.

Related pages in this website

Next, read how to find PAIRS OF FACTORS

Highly Composite Numbers have many factors for their size.

The Cauchy-Schwarz Inequality, which is proved using the quadratic formula

Factoring Whole Numbers, which, if you're not careful, will lead you down a path to Number Theory.

 

The webmaster and author of the Math Help site is Graeme McRae.
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