Did you notice that the terms can be grouped in
2
different ways yet result in the
same
two factors?
Let’s try two more examples that involve the
subtraction
of some
terms.
Example 3
Factor by grouping:
ax + bx
–
ay
–
by
Step 1
ax + bx
–
ay - by
group
terms in pairs
Step 2
= x(a + b)
–
y(a + b)
extract
GCF
from each pair
Step 3
=
(a + b)
(x
–
y)
extract
binomial
common factor
?
Group terms
together that have
a common factor!
NB:
(a + b) = (b + a)
NB
When both terms
have a
negative
constant, factor out a
negative.
This changes
the sign of both terms.

10C
–
P4-5
–
LG
–
Factoring
Page 13 of 35
Remember, there should be
another
grouping of terms that will also work!
Step 1
ax
–
ay + bx
–
by
Step 2
= a(x
–
y) + b(x
–
y)
Step 3
=
(x
–
y)
(a + b)
Did you notice that we do not have to worry about signs with this grouping?
If possible,
when factoring by grouping,
try to group terms so that the third sign is a plus sign
!
One last example before you try some on your own.
Example 4
Factor by grouping:
3x
2
+ 9xy
–
2xy
–
6y
2
if
possible, let’s keep the
third
sign a
plus
sign when grouping!
Step 1
3x
2
–
2xy + 9xy
–
6y
2
group
terms in pairs
Step 2
= x(3x
–
2y) + 3y(3x
–
2y)
extract
GCF
from each pair
Step 3
= (3x
–
2y)(x + 3y)
extract
binomial
common factor
Check these last 2 examples on your own by multiplying the factors.
Using your calculator to verify factoring
To verify whether the factors are correct, follow the procedure:
Step 1:
Enter the original polynomial into “Y1 =”.
Step 2:
Enter the factored version into “Y2 =”.
Step 3:
Execute the Graph command and observe carefully.
Step 4:
If you see two distinct graphs, then the polynomial has been incorrectly
factored. If the calculator graphs one graph on top of the other, then the polynomial has
been graphed correctly.

10C
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P4-5
–
LG
–
Factoring
Page 14 of 35
Practice Exercise
Factor the following polynomials using the
factoring by grouping
method.
1.
2mn + n
2
+ 4m + 2n
6.
bx + b + x + 1
2.
ns + ms + nr + mr
7.
xy
y + 2x
2
3.
mr + ms
–
nr
–
ns
8.
6x
2
3x + 2xy -y
4.
2m
2
+ 4m
–
3m -6
*9.
x
3
+ x
x²
1
5.
abx
2
–
axy
–
bxy + y
2
*10.
5y
3x + 3x²
5xy
Answers to Practice Exercise
1.
2mn + n
2
+ 4m + 2n
= 2mn + n
2
+ 4m + 2n
= n(2m + n) + 2(2m+n)
= (2m + n)(n + 2)
6.
bx + b + x + 1
= bx + b + x + 1
= b(x + 1) + 1(x + 1)
= (x + 1)(b + 1)
2.
ns + ms + nr + mr
= ns + ms + nr + mr
= s(n + m)
+ r(n + m)
= (n + m)(s + r)
7.
xy - y + 2x - 2
= xy - y + 2x - 2
= y(x - 1) + 2(x - 1)
= (x
- 1)(y + 2)
3.
mr + ms - nr - ns
= mr + ms - nr - ns
= m(r + s) - n(r + s)
= (r + s)(m - n)
8.
6x
2
- 3x + 2xy - y
= 6x
2
- 3x + 2xy - y
=3x(2x - 1) + y(2x - 1)
=(2x - 1)(3x + y)
4.
2m
2
+ 4m - 3m - 6
= 2m
2
+ 4m - 3m - 6
= 2m(m + 2)
- 3(m + 2)
= (m + 2)(2m - 3)
9.
x
3
+ x -
x² - 1
= x
3
+ x -
x² - 1
= x(x
2
+1) - 1(x
2
+ 1)
= (x
2
+ 1)(x - 1)
5.
abx
2
- axy - bxy + y
2
= abx
2
- axy - bxy + y
2
= ax(bx - y) - y(bx - y)
= (bx - y)(ax - y)
10.
5y - 3x + 3x² - 5xy
= 3x
2
- 3x - 5xy + 5y (rearrange)
= 3x
2
- 3x - 5xy + 5y
=
3x(x - 1) - 5y(x - 1)
= (x - 1)(3x - 5y)