Jessa Barber
Choose a factoring example,
and follow the steps in order to
solve it. It can be 2, 3 or 4
terms.
Step 1.
How many terms are there?
2
3
4
Step 2.
Is there a GCF?
YES
NO
Step 3.
If there is no GCF, then ‘factor by grouping’.
Ex) pq + pr – sq – sr
a)
Group the terms that have common variables (pq + pr) – (sq – sr)
b)
Remove a GCF from
If the1st term that you are grouping is
the grouped terms
negative, make the whole bracket
negative.
p (q + r) – s (q + r)
d)
* When factoring by grouping, the brackets must be the same, or you
have done something wrong
p (q + r) – s (q + r)
c)
Click, if the brackets are not the same.
Factor out the brackets by removing a GCF from the new set of
terms.
(q + r) (p – s)
NEXT
Step 3.
If there is no GCF, then ‘factor by grouping’.
Ex) pq + pr – sq – sr
a)
Group the terms that have common variables (pq + pr) – (sq – sr)
b)
Remove a GCF from
If the1st term that you are grouping is
the grouped terms
negative, make the whole bracket
negative.
p (q + r) – s (q + r)
d)
* When factoring by grouping, the brackets must be the same, or you
have done something wrong
p (q + r) – s (q + r)
c)
Click, if the brackets are not the same.
Factor out the brackets by removing a GCF from the new set of
terms.
(q + r) (p – s)
NEXT
If brackets are not the same:
Ex) 20x(x - 3) – 4(3 - x) The brackets are
similar, but not exactly the same.
a) Make brackets the same, by factoring out
a (-1)
Ex) 20x(x - 3) – (- 1) 4(x - 3)
= 20x(x-3) + 4(x-3)
Go back
Step 2.
Is there a GCF?
YES
NO
Step 2.
Is there a GCF?
YES
NO
Step 3.
Which form is the trinomial in?
x2 + bx + c
x2 + bxy + cy2
ax2 + bx + c or
ax2 + bxy + cy2
Where ‘a’ does not
equal 1
Step 4.
‘Factoring Quadratic Trinomials’
Where the leading coefficient is not 1, and does not factor out
There are 3 methods which allow you to
solve:
Choose the one you find most helpful:
“OI”
Decomposition
Rumsy’s
Step 5.
“Oi Method” – Guess and check
 Check as you go, but only the inside and outside. FOIL
1.
2.
3.
4.
Ex) 2x2 + 7x + 6 – Nothing factors out
Draw 2 sets of brackets – (
)(
)
Find 2 numbers that multiply to 1st term – (2x )(x
)
Guess and check to find two numbers that multiply to the last
term, but also work with FOIL
3x
(2x + 3) (x + 2)
4x
4x + 3x = 7x
3x2=6
NEXT
Step 5.
Decomposition
Ex) 2x2 + 7x + 6
a)
Multiply 1st number to last number
2 x 6 = 12
b)
Find two numbers that multiply to number in step a, and add to middle
term of trinomial.
3 x 4 = 12
3+4=7
c)
Replace the middle term with the two numbers from step b.
2x2 + 4x + 3x + 6
d)
Factor our a GCF in pairs (factor by grouping)
= (2x2 + 4x) + (3x + 6)
= 2x(x + 2) + 3(x + 2)
e)
Factor out brackets
(x + 2) (2x + 3)
NEXT
Step 5.
Rumsy’s Method
Ex) 2x2 + 7x + 6
•
Multiply 1st number to last number 2 x 6 = 12
•
Replace last number with number in step a. 2x2 + 7x + 12
•
Draw 2 sets of brackets, and put the first term as well as the first variable (x) on
the left of each bracket.
(2x
) (2x
)
•
Find the last 2 numbers by finding two numbers that multiply to equal the last term,
and add to equal the middle term (2x + 3)(2x + 4)
3 x 4 = 12
3+4=6
e)
Divide one or both sets of brackets by a GCF but do not put the GCF outside of
the brackets. (2x + 3)(x + 2)
NEXT
Step 4.
‘Factoring Quadratic Trinomials 2’ – (x2 + bxy + cy2)
Ex) n2 + 10ny + 21y2
• Draw 2 sets of brackets
( )( )
2. The 1st term in each bracket must multiply to get the 1st term in the
trinomial.
(n
)(n
)
4. Find two numbers that multiply to the last term, and add to the second.
Because the last term has a variable2, you must add a variable to these
terms:
7y x 3y = 21y2 and 7y + 3y = 10y2
5. Put those on the right of the brackets
(n + 3y)(n + 7y)
NEXT
Step 4.
‘Factoring Quadratic Trinomials’ – (x2 + bx + c)
Ex) n2 + 10n + 21
• Draw 2 sets of brackets
( )( )
2. The 1st term in each bracket must multiply to get the 1st term in the
trinomial.
(n
)(n
)
3. Find two numbers that multiply to the last term, and add to the second.
7 x 3 = 21
7 + 3 = 10
4. Put those on the right of the brackets
(n + 3)(n + 7)
NEXT
Step 3.
Is it a difference of squares?
Ex) 4x-16 (To be a difference of squares, it
must be a perfect square – perfect square
YES
NO
Step 4.
How to factor a difference of squares:
Ex) 9x2-16y2
a)
Draw 2 sets of brackets (
)(
)
b)
The first term in both sets of brackets, must multiply to get the first
term.
(3x )(3x ) Because it is a perfect square, the first numbers will
be equal to each other.
c)
The second term in each bracket must also multiply to get the
second term.
(3x 4y)(3x 4y)
d)
Put a positive sign in one bracket, and a negative sign in the
other. (This is so that once it is FOILed out, the second term will still be
negative)
(3x + 4y)(3x – 4y)
NEXT
Step 4.
Does it look something like this?
- 4x(x+7) – 3 (x+7) Click, if the
brackets are not
the same.
a)
b)
Remove the GCF. In this case, (x+7) is the GCF
Divide each term by the GCF - (x+7)
= (x + 7)(4x – 3)
NEXT
Step 5.
If brackets are not the same:
Ex) 20x(x-3) – 4(3-x) The brackets are similar, but not
exactly the same.
a) Make brackets the same, by factoring out a (-1)
Ex) 20x(x-3) – (- 1) 4(x-3) = 20x(x-3) + 4(x-3)
NEXT
Step 6.
Does it look something like this?
- 20x(x-3) + 4(x-3)
a) Remove the GCF. In this case, (x-3) is the GCF
b) Divide each term by (x-3)
= (x-3)(20x + 4) This can be simplified
c) To simplify, divide each term in the second bracket by
4, and then put the 4 outside of the brackets.
= 4(x-3)(5x+1)
NEXT
Step 3.
Finding and removing a GCF:
The GCF between terms uses only the common
bases and lowest exponents.
Example: 12ab, 15a2b3  GCF = 3ab
Find the GCF, and put it outside of a set of
brackets. GCF(
)
Then divide each term inside the brackets with the
GCF
NEXT
Step 4.
Now, is your example fully factored?
Ex) Factored: 3ab(x+1)
Not Fully Factored: 3ab(x2+y2)
Factored
Not Factored
Step 5.
Now, is your example fully factored?
Ex) Factored: 3ab(x+1)
Not Fully Factored: 3ab(x2+y2)
Factored
Not Factored
Step 4.
Check that you have factored correctly by
using ‘FOIL’
Step 5/6.
Check that you have factored correctly by
using ‘FOIL’
Step 7.
Check that you have factored correctly by
using ‘FOIL’
DONE!