1.3 Factoring Polynomial and Quadratic Expressions

MATH 1324 – Business College
Lesson 1:
1.3 Factoring
Unit 1 Mathematics Review and Functions

Concepts and Objectives
• Objectives for this section:
• Review factoring polynomial and quadratic expressions
by
• Factoring out the greatest common factor
• Factoring when a = 1
• Factoring when a > 1

Factoring Polynomials
Some basic definitions
polynomial – a mathematical expression in which each
term has an integer exponent
monomial – one term
binomial – two terms
trinomial – three terms
greatest common factor – the greatest number or variable
that are present in each term of the polynomial

Factoring Polynomials
• The process of finding polynomials whose product equals
a given polynomial is called factoring.
• For example, since 4x + 12 = 4x + 3, both 4 and x + 3 are
called factors of 4x + 12.
• A polynomial that cannot be written as a product of two
polynomials of lower degree is a prime polynomial.
• One nice aspect of this process is that it has a built-in
check: whatever factors you come up with, you should be
able to multiply them and get your starting expression.

Factoring Out the GCF
Factor out the greatest common factor from each
polynomial:
• GCF: y2
• GCF: 2t
• GCF: 7m + 1

5 2
9y y
2
6 8 12
x t xt t
 
     
3 2
14 1 28 1 7 1
m m m
    
 
2 3
9 1
y y
 
 
2
3 4 6
2 x
t x
  
     
2
2 1 4 1 1
7 1
m m m
 
    




Factoring Out the GCF (cont.)
We can clean up that last problem just a little more:
     
2
7 1 2 1 4 1 1
m m m
 
    
 
     
2
7 1 2 2 1 4 1 1
m m m m
 
     
 
  2
7 1 2 4 2 4 4 1
m m m m
 
     
 
  
2
7 1 2 3
m m
 

Factoring Trinomials
If you have an expression of the form ax2
+bx + c, you can
use one of the following methods to factor it:
• X-method (a = 1): If a = 1, this is the simplest method to
use. Find two numbers that multiply to c and add up to b.
These two numbers will create your factors.
• Example: Factor x2
‒ 5x ‒ 14.
‒14
‒5
  
2
5 14 7 2
x x x x
    
c
b
–7 2

Factoring Trinomials (cont.)
• If a > 1, I prefer using a method called the Mustang
method: This method is named after the mnemonic “My
Father Drives A Red Mustang”, where the letters stand for:
• If you are solving an equation, you don’t have to bother
moving the denominators; you can just stop at “R”.
M Multiply a and c.
F Find factors using the X method. Set up  .
DA Divide the numeric terms by a if necessary.
R Reduce any fractions.
M
Move any denominators to the front of the
variable.

Factoring Trinomials (cont.)
• Example: Factor
2
5 7 6
x x
 
M Multiply ac
F Find factors:
DA Divide by a
R Reduce fractions
M Move the denominator
‒30
7
  
0 3
1
x x
 
10 3
5 5
x x
  
 
  
  
 
3
2
5
x x
 
 
 
 
  
2 5 3
x x
 
  
5 6 30
 
–3
10

Sidebar: Calculator Shortcut
• If you have a TI-83/84, one way your calculator can help
you find the factors is to do the following:
• In o, set Y1= to ac/X (whatever a and c are)

• In Y2=, go to ½; then select , À, and À. This should put
Y1 in the Y2= line. Then enter Ã„.

• Go to the table (ys). What you’re looking for is a Y2 that
equals b. The values of X and Y1 are your two factors.

Factoring Binomials
• If you are asked to factor a binomial (2 terms), check first
for common factors, then check to see if it fits one of the
following patterns:
• Note: There is no factoring pattern for a sum of squares
(a2
+ b2
) in the real number system.
Difference of
Squares
a2
‒ b2
= a + ba ‒ b
Sum/Diff. of Cubes
  
3 3 2 2
a b a b a ab b
   


Factoring Binomials (cont.)
Examples
• Factor
• Factor
• Factor
2
4 81
x 
3
27
x 
3
3 24
x 
 
2 2
2 9
x
 
3 3
3
x
 
 
3
3 8
x
 
  
2 9 2 9
x x
  
  
2
3 3 9
x x x
 
 
 
3 3
3 2
x
 
  
2
3 2 2 4
x x x
   

Summary
Were the objectives raised at the beginning addressed?
Do you feel that you can
• factor polynomial/quadratic expressions by finding the
GCF
• factor quadratic expressions when a = 1
• factor quadratic expressions when a > 1
Please be sure to fill out the exit ticket and turn it in before
you leave.

For Next Class
• HW: 1.3 Factoring (MyMathLab)
• You do not have a quiz over this section.
Reminder: You may retake this as many times as you like
until Sunday at 11:59 pm.

1.3 Factoring Polynomial and Quadratic Expressions

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