Download Special Cases

Rules of the Signs
Both are positive
Signs are the same
Ax 2
Bx
Bx
C Bx
First sign is positive
Bx
C
Bx
C
C
Signs are the same
Ax 2
Bx
C
Signs are different
Ax 2
Bx
C
Both are negative
Bx
C
Bx
First sign is positive
Ax
Bx
2
C
Signs are different
Bx
C
C
Bx
C
C
-12
3x 2
x
(3x −
)(3x +
4
)
Step #1: Create two sets of parenthesis
Step #2: Put the “Ax” term in each set of parenthesis (you will reduce later)
Step #3: Ask yourself “what two numbers multiplied together give me −12 and
combined give me −1?” (−1 represents the “B” term as it does when we are
working with trinomials when the “A” term is 1.
Answer: +4 and −3 are the two numbers that = −12 when you multiply them,
and also = −1 when you combine them
(3x − 4)(3x + 3)
Step #4: Plug the numbers in the appropriate places; the larger number goes in
the first set of parenthesis (right now).
Step #5: Ask you self which one or both sets of parenthesis you can reduce.
(In this case we mean factor and then throw out the number).
(3x − 4)(x + 1)
Step #6: Check our work.
Do the first terms in the two sets of binomials equal the first term of the
trinomial when multiplied together?
Do the second terms in the binomials equal the last term of the trinomial
3x2
(3x − 4)(x + 1)
x
4
(3x − 4)(x + 1)
One or both
are positive
Ax 2
Positive
means the
Signs will be
the same
Bx
One or both
are negative
C
Negative
means the
signs will be
different
+210
14 x 2
41 x
15
(14x − 35) (14x − 6)
Step #1: Create two sets of parenthesis
Step #2: Put the “Ax” term in each set of parenthesis (you will reduce later)
Step #3: Ask yourself “what two numbers multiplied together give me a
positive 210 and combined give me negative 41?” (−1 represents the “B” term as
it does when we are working with trinomials when the “A” term is 1.
Answer: +4 and −3 are the two numbers that = −12 when you multiply them,
and also = −1 when you combine them
(3x − 4)(3x + 3)
Step #4: Plug the numbers in the appropriate places; the larger number goes in
the first set of parenthesis (right now).
Step #5: Ask you self which one or both sets of parenthesis you can reduce.
(In this case we mean factor and then throw out the number).
(3x − 4)(x + 1)
Step #6: Check our work.
Do the first terms in the two sets of binomials equal the first term of the
trinomial when multiplied together?
Do the second terms in the binomials equal the last term of the trinomial
3x2
(3x − 4)(x + 1)
x
4
(3x − 4)(x + 1)
Factoring Special Cases
The difference of two perfect squares
The difference
x
2
49
x
(
Perfect squares
+
2
49
)(
−
)
Step #1: Write out two sets of parentheses with a
plus in one and a minus in the other
( x + 7 )( x − 7 ) Step #2: Put the square root of the “A” term in front of
the sign and the square root of the “C” term after the
sign.
The perfect square trinomial
Tells you what the sign will be
The 1st and 3rd
terms will
always be
perfect squares
x
2
2 x
1
Perfect squares
( x + 1)
2
Tells you what
the sign will be
The 1st and 3rd
terms will
always be
perfect squares
This will always be positive
because both signs need to
be the same in a perfect
square trinomial
4 x 2 20 x 25
Perfect squares
( 2x − 5)
2