Factoring
Factoring is the process of writing a number (or variable expression) as a product of factors.
Find the factors of:
a) 54
b) 100
Factoring a polynomial means writing the polynomial as a product of other polynomials.
GCF stands for _________________________________________________, the largest monomial
which divides evenly into each term of a polynomial.
Ex1. The GCF of 12a + 16b + 24ab is _________ . We don't include a's or b's in the result because
there is no a in the _____________ term, and no b in the ______________ term. In other words, a and
b are not __________________ to all terms in the polynomial.
Ex2. The GCF of
x 3 + x 5 − x 2 is ______________.
For variables, the GCF is the
_______________ power of all variables that are _________________ to all terms in the polynomial.
Ex3. The GCF of 9 x 3 + 6 x y is
___________.
1
The GCF Method of Factoring
Ex1. Factor: 12a + 16b + 24ab;
(
12 a
3
+
16 b
5
2
+
24 a b
) =
Ex2. Factor:
x +x −x
Ex3. Factor:
9 x3 + 6 x y
Ex4. Factor:
15 x3 y − 10 x 2 y − 5 x y
Ex5: Factor:
3x + 4 y
2
;
the GCF is ____
the GCF is _____
2
2
Factoring by Grouping
Break a 4-term polynomial into 2 groups, GCF each group, and then factor out the common binomial.
Ex1.
Factor: 2 x − 4
+
xy − 2y
common binomial is ___________
factor it out front, leave other pieces
Ex2. Factor:
15 x2 + 6 x + 5 x y 2 + 2 y 2
Important: if the groups are connected by a __________ sign, you need to factor a ________________
out of the second group, changing all the signs to their opposite.
Ex3. Factor:
2a x − 3a y
−
2b x + 3 b y
3
Ex4. Factor:
8 y2 + 4 y
−
6 x y − 3x
Sometimes you need to factor out a 1 or a -1 :
Ex5. Factor:
2
4 x − 2 xy
−
2x + y
And sometimes this type of factoring just doesn't work:
Ex6. Factor:
4 x3 + 8 x2 +
3x − 6
4
Factoring Trinomials of the form x2 + bx + c
Factoring by this method is really "unFOILing". For instance:
FOIL:
( x+ 3)( x+ 4) =
Factor:
=
(x+ ___)( x + ___ )
Ask yourself: what two numbers multiply to _______ and at the same time, add up to ______ .
Ex1. Factor
x 2+11 x +28
What two numbers multiply to _______ and add up to ________ ? Answer:
Ex2. Factor
x 2−6 x−55
What two numbers multiply to _______ and add up to ________ ? Answer:
Ex3. Factor:
36−5 x−x
2
(Rewrite in descending order and factor out -1 first.)
5
Factoring Trinomials of the form ax2 + bx + c using Trial and Error
Again, you need to think "unFOIL".
Ex1. Factor:
5 x2 +11 x +2 = ( ___ x + ___ ) ( ___ x + ___ )
The Firsts (F) must multiply together to give you
, only possible combination: ____________
The Lasts (L) must multiply together to give you 2, only possible combination: _________________
You need to try all combinations of lasts in all orders.
Two possible factorizations: __________________________ and ____________________________
Which one works?
FOIL the middle
(OI) terms.
Outer + Inner
Outer + Inner
Get 11x?
Answer:
Ex2. Factor 2 y 2+5 y −12
Possible Firsts: 2y, y
Possible Lasts: 1, 12; 2, 6; 3, 4 in all orders with different signs.
Possible Combinations, mentally foil middle terms O + I
(2y + 1)(y - 12)
(2y - 1)(y + 12)
(2y + 2)( y - 6)
(2y + 2)( y – 6)
(2y + 3)( y – 4)
(2y - 3)( y + 4)
(2y + 4)( y – 3)
(2y - 4)( y + 3)
(2y + 6)( y – 2)
(2y - 6)( y + 2)
(2y + 12)( y – 1)
(2y - 12)( y + 1)
6
Shortcut 1 - since the original trinomial had no GCF*, neither can its factors!
(2y + 1)(y - 12)
(2y - 1)(y + 12)
(2y + 2)( y - 6)
(2y + 2)( y – 6)
(2y + 3)( y – 4)
(2y - 3)( y + 4)
(2y + 4)( y – 3)
(2y - 4)( y + 3)
(2y + 6)( y – 2)
(2y - 6)( y + 2)
(2y + 12)( y – 1)
(2y - 12)( y + 1)
Shortcut 2 - the O+I (middle terms) of (ax +b)(cx −d) is the opposite of (ax−b)( cx +d)
O+I
opposite O + I
(2y + 1)(y - 12)
(2y - 1)(y + 12)
(2y + 3)( y – 4)
(2y - 3)( y + 4)
We need middle terms to add to ________, because of 2 y 2+5 y−12 . Answer:
Ex3. Factor: 10 x2 +17 x−20 .
(10x +
)(x 1
2
4
5
10
20
)
- 20
- 10
- 5
- 4
- 2
- 1
or
(5x +
)(2x 1
2
4
5
10
20
)
- 20
- 10
- 5
- 4
- 2
- 1
7
Rule out GCF combinations. For instance, (10x + 2)(x - 10) is no good because GCF of (10x+2) = 2.
(10x +
)(x -
)
O+I
(5x +
)(2x -
)
1
- 20
1
- 20
2
- 10
2
- 10
4
- 5
4
- 5
5
- 4
5
- 4
10
- 2
10
- 2
20
- 1
20
- 1
O+I
Which O + I adds up to ______ ?
Ex4.
Factor 6 x 2−23 x+15
8
Difference of Perfect Squares
This one is much easier. Recognize that two "perfect squares" are being subtracted.
General form:
2
2
a −b = (a+b)(a−b)
2
Ex1.
Factor:
Ex2.
Factor 25 − y 6
Ex3.
Factor
4x − 9
... think ...
2
[2 x ] − [3]
2
=
... think ...
t 2 + 64
*** IMPORTANT - The SUM of perfect squares is generally NONFACTORABLE ***
Ex4.
Factor
x 2−8
Overview so far, and Factoring Completely
Generally, you'll need to figure out what type of factoring to apply for a given polynomial. However,
you should ALWAYS try to factor by GCF first.
Ex1. Factor
4 x 2 − 40 x + 100
(Note: this problem would have been tough using the a x 2+ b x+ c method!)
9
Ex2. Factor 2 y 5 − 32 y 3
4
Ex3. Factor
x −16
Ex4. Factor
x 3 − 9 x − 4 x 2 + 36
10
Sums and Differences of Perfect Cubes
Before we start, you need to be aware of two things:
1. What is a perfect cube? Any number or expression that is the result of something else cubed (raised
to the third power).
Examples of perfect cubes:
1, 8, 27, 64, 125 ... because these numbers are 13 , 23 , 3 3 , 4 3 and 53 respectively.
3
6
64 a
3
9
x , x , x ,x
12
... because these numbers are
3
2 3
3 3
4 3
(x) , ( x ) , (x ) and (x )
respectively.
because it's equal to
6
125 x y
9
because it's equal to
(divide exponents by 3).
2. That Multiplying and Factoring "undo" each other.
With this in mind, multiply this expression:
(a + b)(a2 − ab + b 2)
This suggests that ____________ factors into:
__________ times ________________________
Sum and Difference of Perfect Cubes - formulas
a3 +b 3=(a+b)(a2−ab+ b2 )
... sometimes applied as ... (a+ b)(a⋅a − a⋅b + b⋅b)
3
3
2
2
a −b =(a−b)(a + ab+b ) ... sometimes applied as ... (a−b)(a⋅a + a⋅b + b⋅b)
11
x 3+27 ... think ... [x ]3 + [3]3
Ex1. Factor
Formula:
3
3
2
2
a + b = (a + b)(a − ab + b ) = ( a + b)(a⋅a − a⋅b + b⋅b)
Ex2. Factor 1−8 y 3 ... think ...
Formula a3 − b 3 = (a − b)(a2 + ab + b2) = (a − b)(a⋅a + a⋅b + b⋅b)
Factoring Trinomials in Quadratic Form
Sometimes we can apply techniques for one type of problem to other similar problems. For instance,
we know how to factor x 2−8 x−20
2
x −8 x−20 =
If there's symmetry - such as all powers being doubled or tripled, you may be able to factor
x 4−8 x 2−20
Sometimes there's reverse symmetry. The "b" terms increase from a power of 0 to a power of 2.
2
a −8 ab−20 b
2
Check your work (FOIL)!
12
Sometimes it's not meant to be. Notice the y's are not symmetrical:
2
2
x −8 x y −20 y
3
Ex1. Factor:
x 2 y 2−5 x y−66
Ex2. Factor:
28 a + 11 a b + b
2
2
13
Review Problems - factor these polynomials completely:
1.
6 x 3+ 12 x 2−18 x
3.
5.
4
2.
25−x
4 x +81
4.
3 x−2 y−6 x + 4 xy
6 x 2−x−15
6.
6 y−6 y
2
2
5
14
Solving Equations by Factoring
If you multiply two factors together, and the result is zero, then one of the factors must be zero.
Ex1. Given (3x + 2)(x - 7) = 0, then either 3x + 2 = 0, or
equations, you get the solution to the original problem.
3x + 2 = 0
3x = -2
x = -2/3
x - 7 = 0. If you solve both of these
x-7=0
x=7
Solution set: { -2/3, 7 }
Strategy for solving polynomial equations: Set equation equal to 0, factor, set each factor = 0, solve.
Ex2. Solve:
x 2 − 10 x + 24 = 0
Ex3. Solve:
3 x =2 x +1
Ex4. Solve:
9x = 1
2
2
15
Ex5. Solve (x+ 7)(x−1) = 9
Common Mistake
Solution is to expand (x + 7)(x - 1), set the equation equal to zero, and re-factor!
Application Problem - Area
Example: The length of a rectangle is 6 more than twice the width. If the area of the rectangle is 96
sq. feet, what are the dimensions of the rectangle?
Recall the formula for Area of a Rectangle is:
Let the width = x
then, length =
Equation:
16