Download Factoring by Grouping

Math 60
5.2: Factoring by Grouping
Elementary Algebra
In order to understand Factoring by Grouping it helps to review some aspects of polynomial multiplication.
Multiply by using the Distributive Property
Multiply by using the FOIL Method
x  32 x  1
x2 x  1  32 x  1
x  32 x  1
2
x 2  x 6 x 3
2x 2  x  6x  3
F
O
I
L
2
x  5x 3
2
2 x 2  5x  3
F
O I
L
Now we will be taking these multiply problems and reversing the procedure to produce a reverse distributive
process called Factoring by Grouping. Factoring by Grouping is used exclusively to factor 4–termed
polynomials, where a 4–termed polynomial expression is rewritten as a product of two binomial factors.
Factor by Grouping:
Many students consider this
a "reverse distributive process".
Practice Problems:
Factor by Grouping.
You can check your work on any factoring
problem by taking your final answer and
multiplying back using an appropriate
technique (see §4.5) to verify the original
polynomial.
2 x2  x  6 x  3
 2 x2  x  6x  3

2
  2 x  x    6 x  3

 x  2 x  1  3  2 x  1

  2 x  1 x  3
  x  3 2 x  1

For “Factoring by Grouping” problems the
easiest way to check your work is by
FOIL-ing.
For the last two terms of the 4–termed polynomial…
 Lead on a NEGATIVE  Factor out a NEGATIVE.
 Lead on a POSITIVE  Factor out a POSITIVE.
1.
x 2  x  3x  3
2.
9w 2  6w  6 w  4
3.
12 x2  42 x 10 x  35
4.
x2  3xy  4 xy  12 y 2
5.
y3  3 y  2 y 2  6
6.
y3  3 y  2 y 2  6
(put in descending order first)
Whenever you are asked to factor polynomials the expectation is that this process is done completely.
In other words, when we write our factorization we need to verify that each polynomial in the factorization is
non-factorable (or prime). Sometimes it may be necessary to factor more than once to produce this “complete”
factorization.
Factor completely by Grouping.
7.
3z 4  3z 3  7 z 3  7 z 2
3z 4  3z 3  7 z 3  7 z 2
8.
9 x 3  6 x 2  45x 2  30 x
9 x 3  6 x 2  45x 2  30 x
Sometimes the terms in a 4–termed polynomial are not in the proper order to make Factoring by Grouping
possible. If the 4–termed polynomial is factorable, then it is simply a matter of producing the correct
arrangement of terms, so that we can then use Factoring by Grouping. Not all arrangements will allow for
Factoring by Grouping; however, of those that work, the answer will all be algebraically equivalent regardless
of the arrangement of terms. (Hint: Review Problems 5 and 6 on the previous page.)
Factor by Grouping.
9.
ax 10  5x  2a
Answers:
1.
4.
7.
10.
ac  bd  ad  bc
10.
For Chapter 5: Where appropriate the POSITIVE factor has been listed first.
Remember, that since multiplication is commutative:  x  3 x  4    x  4  x  3
 x  3 x  1
 x  4 y  x  3 y 
z 2  z  1 3z  7 
 x  b  c  d 
2.
5.
8.
 3w  2
 y  2   y 2  3
3x  3x  2  x  5
2
3.
6.
9.
 2 x  7  6 x  5
 y  2   y 2  3
 x  2 a  5