Download FACTORING This is probably one of the most important skill

FACTORING
This is probably one of the most important skill-building assignments that you will complete this semester.
Granted, factoring is never used in real life, but you will never be able to escape it in mathematics all
the way through calculus. Thus, you are expected to master all different forms of factoring or the
remainder of this course will most likely become a living nightmare for you. Moreover, you will be
expected to factor with speed. For what it’s worth, recognizing patterns is essential to successful
factoring, and that is one critical thinking skill that you can apply in many real life situations.
Most important, factoring is one of the only ways to solve a quadratic equation.
Prime Factorization vs. Factors
Factoring with numbers allows us to see how we might be able to form them by multiplication. You
should be able
to tell the difference between listing the factors of a number of writing it’s prime-facto
Factored Form
Remember factors are just quantities being multiplied.
Factor · factor · factor = product
Factors can be simple as in
8 · 14 = 112
of they can be complex as in
(2(3) + 2/3)(2(3)-2/3)=36-4/9=355/9
In order to factor you must be able to tell when an expression is in factored form and when it is in
“simplest form” (x+3)(2x+4) ( Factored Form)
2x1+10x+12 (Simplest form)
For each expression below tell whether it is in factored or simplest form.
1. (x + 4)(2x − 5)
Factored form
2. (3x + 2) (4x + 3)
Factored form
3. X2− 3x + 10
Simplest form
4. (2x + 5)(9x − 4)
Factored form
5. 4x + 2
Solving Equations vs. Factoring
Many exercises that you have completed asked you to “factor” an expression. Here’s an example.
Being asked to solve an equation for one variable requires that you isolate that variable and express it’s
value. For example, in the equation 2x + 5 = 13 the value of x must be 4. So when you are asked to solve
2x + 5 = 13, you answer simply x = 4. With more complex equations, “solving” still requires this. Here’s an
example of an equation that can be solved by factoring
Factoring Strategies
Greatest Common Factor (GCF)
Some expressions can be completely factored by simply factoring out the Greatest Common Factor.
For example 4x + 8 can be factored to 4(x + 2), which cannot be further factored.
10x3 + 5x + 20x2 can be factored by the GCF of the terms 10x3, 5x, and 20x2 which is 5x.
10x3 + 5x + 20x2 = 5x(2x2 + 1 + 4x) and that cannot be factored any further. Noticing that the terms of an
expression can be divided by a common factor is an essential factoring skill.
THIS STEP SHOULD ALWAYS BE DONE FIRST! ALWAYS ALWAYS ALWAYS!
Factor each expression
1. 12x + 20
GCF: 4; 4(3x+5)
2. 3x2 + 18x3 + 23x4
GCF: x2; x2(3+18x+23x2)
3. 5x + 10xy + 25x2y2
GCF: 5x; 5x(1 +2y+5xy2)
Factoring by Grouping
We can factor expressions with four terms by grouping. See how to factor x2 + 4x + 15x + 60
x2 + 4x + 15x + 60 = (x2 + 4x) + (15x + 60)
= x(x + 4) + 15(x + 4) (Notice how (x+4) is now the GCF of each term)
= (x + 4)(x + 15)
(“Factor out GCF”)
Factor
1. x2 + 5x + 6x + 30
8x
x(x+5) +6(x+5)
x
(x+5)(x+6)
2. 12x2 + 39x + 28x + 91
3. 20x3 + 5x2 – 32x2 −
3x(4x+13) + 7(4x+13)
FIRST factor out GCF:
(3x+7)(4x+13)
x(20x2+5x-32x-8)
x(5x(4x+1)-8(4x+1)
x(5x-8)(4x+1)
Factoring using Patterns
Recognizing patterns with your quadratic expressions/equations will make factoring very easy. Knowing
“how” to factor a certain expression based on its terms will make factoring fun. I swear!
Difference of Two Squares (for Quadratic Expressions)
What to look for:
Binomial
both terms are perfect squares and they are being subtracted
a2 – b2 = (a – b)(a + b)
Why is this called the difference of squares? Because you are subtracting (DIFFERENCE) TWO Perfect
SQUARE terms. See! Math DOES make sense!
To find “a” and “b” you need to find the square root of a2 and the square root of b2 When you are
solving an equation and you have to take the square root of something, you always get a ± answer. For
using the difference of squares theorem you only need to worry about your positive answer.
Ex. x2 – 9
Ex. a2 – 49 =
a2 = x2 therefore a= x 2 =x
a2 = a2 therefore a= a 2 =a
b2 = 9 therefore b=
9 3
b2 = 49 therefore b=
49  7
plugging a and b into the formula (a+b) (a-b)
the factored form of x2 – 9 is
plugging a and b into the formula (a+b) (a-b)
the factored form of a2 –4 9 is
(x+3)(x-3)
(a+7)(a-7)
‘
Ex. 25x2 – 16y2
Ex. 1 – 49y2
a2 = 25x2 therefore a= 25x 2 =5x
a2 = 1 therefore a= 1 =1
b2 =16y2 therefore b= 16 y 2  4 y
b2 = 49y2 therefore b= 49 y 2  7 y
plugging a and b into the formula (a+b) (a-b)
the factored form of25x2 – 16y2 is
plugging a and b into the formula (a+b) (a-b)
the factored form of 1 – 49y2 is
(5x+4y)(5x-4y)
(1+7y)(1-7y)
Remember! With any factoring problem, ALWAYS look for a GCF first!. Something may not
appear to be a difference of two squares, but it really is.
Ex. 3x2 – 75 =
GCF: 3
Factoring that out.. 3(x2-25)
(x2-25) is a difference of two squares
Ex. 20x3 – 5x =
GCF: 5x
Factoring that out.. 5x(4x2-1)
(4x2-1) is a difference of two squares
Now a2 = x2 therefore a= x 2 =x
Now a2 = x2 therefore a= 4x 2 =2x
b2 =25 therefore b= 25  5
plugging a and b into the formula (a+b) (a-b) AND
remembering that there is a 3 on the outside of
everything. IT DOESN’t JUST GO AWAY! You get
3(x+5)(x-5)
b2 =1 therefore b= 1  1
plugging a and b into the formula (a+b) (a-b) AND
remembering that there is a 5x on the outside of
everything. IT DOESN’t JUST GO AWAY! You get
5x(2x+1)(2x-1)
Perfect Square Trinomials
What to look for:
The first and last terms are perfect squares.
The middle term is two times one factor from the first term and one factor from the last term.
How it factors: a2+2ab+b2 = (a+b)(a+b)= (a+b)2
Ex. x2 +6x+9
Ex. x2 +10x+25
“a” and “c” are both perfect squares
6 is two times 1 and 3 (factors from a and c)
“a” and “c” are both perfect squares
10 is two times 1 and 5 (factors from a and c)
So using your equation you would get
So using your equation you would get
(x+3)(x+3)
(x+5)(x+5)
Ex. 4x2 +4x+2
Ex. 9x2-12x+4
“a” and “c” are both perfect squares
4 is two times 1 and 2 (factors from a and c)
“a” and “c” are both perfect squares
12 is two times 2 and 3 (factors from a and c)
So using your equation you would get
So using your equation you would get
(2x+1)(2x+1)
(3x-2)(3x-2)
THE “X” Method
For ANY factorable QUADRATIC expression the “X” method ALWAYS works. Its takes a bit
longer and you have to remember the
ax2+bx+x
process, but it ALWAYS works. ALWAYS. I HEART
a*c
the “X” Method
*
____
_____
a
a
+
b
The left and right of the x will contain two factors
of the product of ac whose sum is equal to b
Examples
x2-5x-6
x2+21x+110
4x2 + 12x + 9
3x2 + 10x + 3
Higher Degree Polynomials
We will factor higher degree polynomials more next semester, but some cubic expressions can fall under
the following two categories:
What to look for:
Binomial
Two Perfect Cubes
Difference of two cubes
ab+b2)
(a3-b3) =(a-b)(a2+ab+b2)
Sum of two cubes
(a3+b3) =(a+b)(a2-
Examples:
64a3+125
8x3 -125
y3+ 64
a= 4a; b=5
a=3x; b=5
a=y; b=4
(4a+5)(16a2-20a+25)
(3x-5)(9x2-15x+25)
(y+4)(y2-4a+16)