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This lesson will review the process of factoring,
which is used when solving
equations and simplifying rational expressions.
To factor polynomial expressions, there are several
approaches that can be used to simplify the process. While all of these
approaches are
not used for each problem, it is best to examine your expression for
the possible existence of these situations. Ask yourself the
following questions:
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Are there Common Factors?
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Factor out
the Greatest Common Factor (GCF) of the expression, if one
exists. This will make it simpler to factor the
remaining expression.
Take care NOT to drop
this GCF, as it is still part of the expression's answer.
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Does the expression
have only 2 terms? |
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If it does, is the
expression a DIFFERENCE of PERFECT SQUARES?
If so, you should be able to write the expression as a product of the sum
and difference of the square roots of the terms.
Sometimes, as in Example
2 below, it is best to write the terms in square notation so
you can see what the terms will be
in factored form. Be sure to use parentheses!
This process is also
called Factoring with DOTS (Difference of Two Squares).
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Check out this problem,
which combines
both of the above concepts. |
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Factor completely:
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Problem has two
subtracted terms
and it has a common factor. |
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Factor out the GCF first
(making the next step easier)
Now, factor the
Difference of Squares expression. |
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Notice that another
Difference of Squares shows up!
Factor again |
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Now, that's
factored!!!! |
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Does the expression
have exactly 3 terms? |
If yes, then the expression may factor
into the product of two binomials. One way to solve
this type of problem is to use
trial and error, keeping certain "hints" in mind.
Hints:
With the trinomial arranged in proper order
(highest to lowest powers):
• if the leading coefficient is 1, you are
looking for two numbers that multiply to the
last term and add to the coefficient of the
middle term.
• if the leading coefficient is not 1, you
will have to look more carefully to find the
answer. See
Factoring Trinomials ( )
- Set Up, Guess and Check Method and
Factoring by
Grouping Method.
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For the examples below, use the hint above
for factoring when the leading coefficient is 1, and the trial and error
(guess and check) method when the leading coefficient is 2.
Always check your work by multiplying the
binomials to see if your center term matches the original
problem.
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Special trinomial:
Perfect Square |
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Consider what happens when
a binomial is squared:
 
where the center term is twice the
product of a and b.
If you can recognize
this pattern, it is very easy to factor a trinomial that is
the perfect square of a binomial. |
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Does the first term
have a Negative Coefficient? |
If yes, then factor out the negative sign first,
using the common factor method at the
top of this page.
Remember, if the leading term has a
coefficient of (- 1), and there are
NO other common terms,
then the GCF is = -1. |
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| Check out this
Combo! |
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Factor completely:
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This problem has a
leading negative coefficient,
and has common factors. |
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Factor out the GCF. |
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Factor the
perfect square trinomial. |
or
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Express as a
perfect square. |
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