GCF/Factor by Grouping

Factoring Binomials

Factoring Trinomials

Solving Eq by Factoring

Factoring Word Problems

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About Intermediate Algebra

These pages are dedicated for topics covered in an intermediate algebra course.

Greatest Common Factor and Factor by Grouping

Two main concepts you need to understand are Greatest Common Factor and Factor by Grouping.


Greatest Common Factor: The greatest common factor, or GCF, is the greatest factor that divides two numbers. - http://www.math.com/


Factor By Grouping:

  • Remove the GCF if possible

  • Group terms with common factors

  • Remove the GCF of each group

  • Use the distributive law to rewrite the expression


-http://faculty.stcc.edu/zee/factorin1.htm




Factoring Polynomials

Basic idea: when we factor, when look for common factor(s) to take out of each term

To check to see if you factored correctly redistribute the factored terms and you should get back what you started with.


Lets start factoring.


Back to basic

4ab + 3ac

How many terms do you see?

There are two terms 4ab and 3ac. Aterm can be made of multiple variables and numbers.


What does each have in common? 'a'

Once you find the common term take it out.

i.e. a(4b + 3c)

You can check by distributing the 'a'.


A more complex example.

2ab + 4bc

What does each term share in common? --- 2b.

Factor out the 2b and you get ...

2b(a + 2c)

If you wish, check by redistributing the 2b.


Look at another exmple.

What does each have in common or what is the GCF? ---- 5


Write GCF outside parenthesis

5( )

Divde each term by the GCF




Look at this example

What does each have in common or what is the GCF? ---- 12

What is your answer?




Try this one

Is there any variables or constants that all terms share? ----

So factor out the

what is left on the inside of parenthesis

easy enough?


Try another example

10ab – 3bc + 7cd

What can I do here?

Do these three terms share any common factor? --- no

Therefore this polynomial is prime.



Analyze this one

z(2x+5) + (2x+5)

When there is no coefficient on the outside of parenthesis as in (2x+5) there is an understood 1 as the coefficient. ie. (2x + 5) = 1(2x+5)

Both terms share a 2x+5 in common

write the common term in one set of parenthesis. write what was not in common in another set of parenthesis. i.e. (common)(not in common)

(2x+5)(z+1)

And this is how you factor a polynomial.



Analyze this one

(common)(not in common)

Each term in the second set of parenthesis share a common factorof 11k. Factor out thie 11k.

11k(3k-8)(k+3)




Analyze another one

Do the two terms have any thing in common? --- Yes

(4y+1)

Factor out a (4y+1)

Look at the second set of parenthesis. Is there anything in common? ---- 13x

Factor out th 13 x and put it out infront

13x(4y+1)(3 +x)

since addition is communative (3 + x) = (x + 3)

13x(4y+1)(x+3)




Factor by Grouping

Factor by grouping involve 4 terms.

You start by taking your four terms and putting them in two group make sure each group has something in common.


you do not have to group it like mine.


I created two groups. In the the first group there is a common factor of 4. In the second group there is a common factor of 3n.

4(n – 5m)+3n(5m – n)

You can rearage a binomial with a minus sign by switching the positions and taking out a negative

4(n – 5m) – 3n(n – 5m)

I create two sets of parenthesis. The first set is what is in common and the second what is not in common.

(common)(not in common)


(n – 5m)(4 – 3n)


Lets look at another example

make two groups

The first group has a common factor of 3x and the second group has a common factor of 15

3x(3x – y) + 15(3x – y)

(common)(not in common)

(3x – y)(3x+15)

If you look closely you can see that the second term can be factored more.

What can be factored out of the second term? --- 3

3(x+5)(3x-y)

are we done?

We know we are done when we can not factor any more.


Try this one

There are two terms

What does each term have in common? --

what is left on the inside?

remember laws of exponents

so then

You can always check by distributing


Lets try another problem.

What do they each have in common?

They each have an Once again use the laws of exponents.



Below is a Video about GCF

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