<< Chapter < Page Chapter >> Page >
This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. Factoring is an essential skill for success in algebra and higher level mathematics courses. Therefore, we have taken great care in developing the student's understanding of the factorization process. The technique is consistently illustrated by displaying an empty set of parentheses and describing the thought process used to discover the terms that are to be placed inside the parentheses.The factoring scheme for special products is presented with both verbal and symbolic descriptions, since not all students can interpret symbolic descriptions alone. Two techniques, the standard "trial and error" method, and the "collect and discard" method (a method similar to the "ac" method), are presented for factoring trinomials with leading coefficients different from 1. Objectives of this module: be able to factor trinomials with leading coefficient 1, become familiar with some factoring hints.

Overview

  • Method
  • Factoring Hints

Method

Let’s consider the product of the two binomials ( x 4 ) and ( x 7 ) .

The product of two binomials, x plus four and x plus seven, is equal to x squared plus seven x plus four x plus twenty eight, which is simplified to x squared plus eleven x plus twenty eight. The FOIL method is shown by arrows from the first binomial to the second binomial in the product.

Notice that the first term in the resulting trinomial comes from the product of the first terms in the binomials: x x = x 2 . The last term in the trinomial comes from the product of the last terms in the binomials: 4 7 = 28 . The middle term comes from the addition of the outer and inner products: 7 x 4 x 11 x . Also, notice that the coefficient of the middle term is exactly the sum of the last terms in the binomials: 4 7 11 .

The problem we’re interested in is that given a trinomial, how can we find the factors? When the leading coefficient (the coefficient of the quadratic term) is 1, the observations we made above lead us to the following method of factoring.

    Method of factoring

  1. Write two sets of parentheses: ( ) ( ) .
  2. Place a binomial into each set of parentheses. The first term of each binomial is a factor of the first term of the trinomial.
  3. Determine the second terms of the binomials by determining the factors of the third term that when added together yield the coefficient of the middle term.

Sample set a

Factor the following trinomials.

x 2 + 5 x + 6

  1. Write two sets of parentheses: ( ) ( ) .
  2. Place the factors of x 2 into the first position of each set of parentheses:

    ( x ) ( x )

  3. The third term of the trinomial is 6. We seek two numbers whose
    (a) product is 6 and
    (b) sum is 5.
    The required numbers are 3 and 2. Place + 3 and + 2 into the parentheses.

    x 2 + 5 x + 6 = ( x + 3 ) ( x + 2 )

    The factorization is complete. We’ll check to be sure.

    ( x + 3 ) ( x + 2 ) = x 2 + 2 x + 3 x + 6 = x 2 + 5 x + 6

Got questions? Get instant answers now!

y 2 2 y 24

  1. Write two sets of parentheses: ( ) ( ) .
  2. Place the factors of y 2 into the first position of each set of parentheses:

    ( y ) ( y )

  3. The third term of the trinomial is 24 . We seek two numbers whose
    (a) product is 24 and
    (b) sum is 2 .
    The required numbers are 6 and 4 . Place 6 and + 4 into the parentheses.

    y 2 2 y 24 = ( y 6 ) ( y + 4 )

    The factorization is complete. We’ll check to be sure.

    ( y 6 ) ( y + 4 ) = y 2 + 4 y 6 y 24 = y 2 2 y 24

Notice that the other combinations of the factors of 24 (some of which are 2 , 12 ; 3 , 8 ; and 4 , 6 ) do not work. For example,

( y 2 ) ( y + 12 ) = y 2 + 10 y 24 ( y + 3 ) ( y 8 ) = y 2 5 y 24 ( y 4 ) ( y + 6 ) = y 2 + 2 y 24

In all of these equations, the middle terms are incorrect.

Got questions? Get instant answers now!

a 2 11 a + 30

  1. Write two sets of parentheses: ( ) ( ) .
  2. Place the factors of a 2 into the first position of each set of parentheses:

    ( a ) ( a )

  3. The third term of the trinomial is + 30 . We seek two numbers whose
    (a) product is 30 and
    (b) sum is 11 .
    The required numbers are 5 and 6 . Place 5 and 6 into the parentheses.

    a 2 11 a + 30 = ( a 5 ) ( a 6 )

    The factorization is complete. We’ll check to be sure.

    ( a 5 ) ( a 6 ) = a 2 6 a 5 a + 30 = a 2 11 a + 30

Got questions? Get instant answers now!

3 x 2 15 x 42

Before we begin, let’s recall the most basic rule of factoring: factor out common monomial factors first . Notice that 3 is the greatest common monomial factor of every term. Factor out 3.

3 x 2 15 x 42 = 3 ( x 2 5 x 14 )

Now we can continue.

  1. Write two sets of parentheses: 3 ( ) ( ) .
  2. Place the factors of x 2 into the first position of each set of parentheses:

    3 ( x ) ( x )

  3. The third term of the trinomial is 14 . We seek two numbers whose
    (a) product is 14 and
    (b) sum is 5 .
    The required numbers are 7 and 2 . Place 7 and + 2 into the parentheses.

    3 x 2 15 x 42 = 3 ( x 7 ) ( x + 2 )

    The factorization is complete. We’ll check to be sure.

    3 ( x 7 ) ( x + 2 ) = 3 ( x 2 + 2 x 7 x 14 ) = 3 ( x 2 5 x 14 ) = 3 x 2 15 x 42

Got questions? Get instant answers now!

Practice set a

Factor, if possible, the following trinomials.

k 2 8 k 15

( k + 3 ) ( k + 5 )

Got questions? Get instant answers now!

y 2 7 y 30

( y + 10 ) ( y 3 )

Got questions? Get instant answers now!

m 2 10 m 24

( m + 6 ) ( m + 4 )

Got questions? Get instant answers now!

m 2 10 m 16

( m 8 ) ( m 2 )

Got questions? Get instant answers now!

Factoring hints

Factoring trinomials may take some practice, but with time and experience, you will be able to factor much more quickly.

There are some clues that are helpful in determining the factors of the third term that when added yield the coefficient of the middle term.

Factoring hints

Look at the sign of the last term :
  1. If the sign is positive, we know that the two factors must have the same sign, since ( + ) ( + ) = ( + ) and ( ) ( ) = ( + ) . The two factors will have the same sign as the sign of the middle term.
  2. If the sign is negative, we know that two factors must have opposite signs, since ( + ) ( ) = ( ) and ( ) ( + ) = ( ) .

Sample set b

Factor x 2 7 x + 12 .

  1. Write two sets of parentheses: ( ) ( ) .
  2. The third term of the trinomial is + 12 . The sign is positive, so the two factors of 12 we are looking for must have the same sign. They will have the sign of the middle term. The sign of the middle term is negative, so both factors of 12 are negative. They are 12 and 1 , 6 and 2 , or 4 and 3 . Only the factors 4 and 3 add to 7 , so 4 and 3 are the proper factors of 12 to be used.

    x 2 7 x + 12 = ( x 4 ) ( x 3 )

Got questions? Get instant answers now!

Practice set b

Factor, if possible, the following trinomials.

4 k 2 32 k 28

4 ( k + 7 ) ( k + 1 )

Got questions? Get instant answers now!

3 y 4 24 y 3 36 y 2

3 y 2 ( y + 2 ) ( y + 6 )

Got questions? Get instant answers now!

x 2 x y 6 y 2

( x + 2 y ) ( x 3 y )

Got questions? Get instant answers now!

5 a 5 b 10 a 4 b 2 15 a 3 b 3

5 a 3 b ( a + 3 b ) ( a b )

Got questions? Get instant answers now!

Exercises

For the following problems, factor the trinomials when possible.

x 2 + 4 x + 3

( x + 3 ) ( x + 1 )

Got questions? Get instant answers now!

x 2 + 7 x + 12

( x + 3 ) ( x + 4 )

Got questions? Get instant answers now!

y 2 + 8 y + 12

( y + 6 ) ( y + 2 )

Got questions? Get instant answers now!

y 2 5 y + 4

( y 4 ) ( y 1 )

Got questions? Get instant answers now!

a 2 + 3a 4

( a + 4 ) ( a 1 )

Got questions? Get instant answers now!

x 2 - 4x 21

( x 7 ) ( x + 3 )

Got questions? Get instant answers now!

y 2 + 10 y + 16

( y + 8 ) ( y + 2 )

Got questions? Get instant answers now!

y 2 - 8 y + 7

( y 7 ) ( y 1 )

Got questions? Get instant answers now!

a 2 + a 30

( a + 6 ) ( a 5 )

Got questions? Get instant answers now!

a 2 12 a + 20

( a 10 ) ( a 2 )

Got questions? Get instant answers now!

x 2 + 13 x + 42

( x + 6 ) ( x + 7 )

Got questions? Get instant answers now!

x 2 + 13 x + 40

( x + 5 ) ( x + 8 )

Got questions? Get instant answers now!

b 2 + 15 b + 56

( b + 8 ) ( b + 7 )

Got questions? Get instant answers now!

3 a 2 + 24 a + 36
( Hint: Always search for a common factor.)

Got questions? Get instant answers now!

4 x 2 + 12 x + 8

4 ( x + 2 ) ( x + 1 )

Got questions? Get instant answers now!

5 y 2 70 y + 440

5 ( y 2 14 y + 88 )

Got questions? Get instant answers now!

x 3 + 6 x 2 + 8 x

x ( x + 4 ) ( x + 2 )

Got questions? Get instant answers now!

x 4 + 9 x 3 + 14 x 2

x 2 ( x + 7 ) ( x + 2 )

Got questions? Get instant answers now!

4 a 3 40 a 2 + 84 a

4 a ( a 7 ) ( a 3 )

Got questions? Get instant answers now!

2 y 2 n 2 10 y 2 n 48 y 2

2 y 2 ( n 8 ) ( n + 3 )

Got questions? Get instant answers now!

4 x 4 42 x 3 + 144 x 2

Got questions? Get instant answers now!

y 5 + 13 y 4 + 42 y 3

y 3 ( y + 6 ) ( y + 7 )

Got questions? Get instant answers now!

4 x 2 a 6 48 x 2 a 5 + 252 x 2 a 4

Got questions? Get instant answers now!

Exercises for review

( [link] ) Factor 6 x y 2 a x 3 a y a 2 .

( 2 x a ) ( 3 y + a )

Got questions? Get instant answers now!

( [link] ) Factor 4 x 2 17 x 15 .

( 4 x 3 ) ( x + 5 )

Got questions? Get instant answers now!

Questions & Answers

the diagram of the digestive system
Assiatu Reply
How does twins formed
William Reply
They formed in two ways first when one sperm and one egg are splited by mitosis or two sperm and two eggs join together
Oluwatobi
what is genetics
Josephine Reply
Genetics is the study of heredity
Misack
how does twins formed?
Misack
What is manual
Hassan Reply
discuss biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles
Joseph Reply
what is biology
Yousuf Reply
the study of living organisms and their interactions with one another and their environments
AI-Robot
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
Joseph Reply
what is the blood cells
Shaker Reply
list any five characteristics of the blood cells
Shaker
lack electricity and its more savely than electronic microscope because its naturally by using of light
Abdullahi Reply
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
Ramadan
What is classification
ISCONT Reply
is organisms that are similar into groups called tara
Yamosa
in what situation (s) would be the use of a scanning electron microscope be ideal and why?
Kenna Reply
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
Hilary
cell is the building block of life.
Condoleezza Reply
what is cell divisoin?
Aron Reply
Diversity of living thing
ISCONT
what is cell division
Aron Reply
Cell division is the process by which a single cell divides into two or more daughter cells. It is a fundamental process in all living organisms and is essential for growth, development, and reproduction. Cell division can occur through either mitosis or meiosis.
AI-Robot
What is life?
Allison Reply
life is defined as any system capable of performing functions such as eating, metabolizing,excreting,breathing,moving,Growing,reproducing,and responding to external stimuli.
Mohamed
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask