Factoring Polynomials by Grouping

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Factoring Polynomials by Grouping
10.8
Goal 1 Factor GCF Goal 2 Factor by
Grouping (reverse distributive property)
2
 
The GCF for a polynomial is the largest monomial
that divides (is a factor of) each term of the
polynomial.
 
   
Step 1  Identify the GCF of the polynomial.
Step 2   Divide the GCF out of every term of the
polynomial.
     
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Factor out the GCF
 
This looks a little different because our GCF is
a binomial. We treat it in the same manner that
we do a monomial GCF.
 
Note that this is not in factored form because of
the plus sign we have before the 5 in the
problem.  To be in factored form, it must be
written as a product of factors.
Step 1  Identify the GCF of the polynomial.
 
This time it isn't a monomial but a binomial that
we have in common.  Our GCF is (3x 1).
 
 
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Step 2  Divide the GCF out of every term of the
polynomial.
Divide (3x-1) out of both parts
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In some cases there is not a GCF for ALL the
terms in a polynomial.  If you have four or more
terms with no GCF, then try factoring by grouping.
Step 1 Group the first two terms together and
then the last two terms together.
  Step 2 Factor out a GCF from each separate
binomial.
Step 3 Factor out the common binomial.
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  Factor by grouping 
   
Note how there is not a GCF for ALL the terms. 
So lets go ahead and factor this by grouping.
 
 
 
 
     
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Step 3 Factor out the common binomial.
Divide (x 3) out of both parts
Note that if we multiply our answer out, we do
get the original polynomial.
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  Factor by grouping 
 
There is not a GCF for ALL the terms.  So factor
this by grouping.
 
 
Be careful.  When the first term of the second
group of two has a minus sign in front of it, you
want to put the minus in front of the second (  
).  When you do this you need to change the sign
of BOTH terms of the second (  ) as shown above.
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Step 2 Factor out a GCF from each separate
binomial.
 
Factor out a  7x2 from the 1st (  ) Nothing to
factor out from the 2nd (  ) 
 
 
 
Note that if we multiply our answer out that we
do get the original polynomial.
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  Factor by grouping 
2x3 2x2 4x 4
 
 
 
  Factor by grouping 
 
6x3 2x2 - 9x - 3
 
 
 
11
 
  Factor by grouping 
3x3 6x2 9x 18
 
 
 
  Factor by grouping 
 
6x4 2x3 - 9x2 3x
 
 
 
12
 
  Factor by grouping 
5x3 - 10x2 - 4x 8
 
Homework
 
 
 
p. 629, 15-26, 37-39, 41, 43, 44, 46, 70-82e