"The principal goal of education is to create men who are capable of doing new things,
not simply of repeating what other generations have done." --- Jean Piaget

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Tip 1: Use Study Groups

Tip 2: Use Video Lectures

Tip 3: Explain material
to others

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.