<< Chapter < Page Chapter >> Page >

Factor: r 2 3 r 40 .

( r + 5 ) ( r 8 )

Got questions? Get instant answers now!

Factor: s 2 3 s 10 .

( s + 2 ) ( s 5 )

Got questions? Get instant answers now!

Some trinomials are prime. The only way to be certain a trinomial is prime is to list all the possibilities and show that none of them work.

Factor: y 2 6 y + 15 .

Solution

y 2 6 y + 15 Factors will be two binomials with first ( y ) ( y ) terms y .

Factors of 15 Sum of factors
−1 , −15 −1 + ( −15 ) = −16
−3 , −5 −3 + ( −5 ) = −8

As shown in the table, none of the factors add to −6 ; therefore, the expression is prime.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Factor: m 2 + 4 m + 18 .

prime

Got questions? Get instant answers now!

Factor: n 2 10 n + 12 .

prime

Got questions? Get instant answers now!

Factor: 2 x + x 2 48 .

Solution

2 x + x 2 48 First we put the terms in decreasing degree order. x 2 + 2 x 48 Factors will be two binomials with first terms x . ( x ) ( x )

As shown in the table, you can use −6 , 8 as the last terms of the binomials.

( x 6 ) ( x + 8 )
Factors of −48 Sum of factors
−1 , 48 −1 + 48 = 47
−2 , 24
−3 , 16
−4 , 12
−6 , 8
−2 + 24 = 22
−3 + 16 = 13
−4 + 12 = 8
−6 + 8 = 2

Check.

( x 6 ) ( x + 8 )

x 2 6 q + 8 q 48

x 2 + 2 x 48

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Factor: 9 m + m 2 + 18 .

( m + 3 ) ( m + 6 )

Got questions? Get instant answers now!

Factor: −7 n + 12 + n 2 .

( n 3 ) ( n 4 )

Got questions? Get instant answers now!

Let’s summarize the method we just developed to factor trinomials of the form x 2 + b x + c .

Factor trinomials.

When we factor a trinomial, we look at the signs of its terms first to determine the signs of the binomial factors.

x 2 + b x + c ( x + m ) ( x + n )

When c is positive, m and n have the same sign.

b positive b negative m , n positive m , n negative x 2 + 5 x + 6 x 2 6 x + 8 ( x + 2 ) ( x + 3 ) ( x 4 ) ( x 2 ) same signs same signs

When c is negative, m and n have opposite signs.

x 2 + x 12 x 2 2 x 15 ( x + 4 ) ( x 3 ) ( x 5 ) ( x + 3 ) opposite signs opposite signs

Notice that, in the case when m and n have opposite signs, the sign of the one with the larger absolute value matches the sign of b .

Factor trinomials of the form x 2 + bxy + cy 2

Sometimes you’ll need to factor trinomials of the form x 2 + b x y + c y 2 with two variables, such as x 2 + 12 x y + 36 y 2 . The first term, x 2 , is the product of the first terms of the binomial factors, x · x . The y 2 in the last term means that the second terms of the binomial factors must each contain y . To get the coefficients b and c , you use the same process summarized in the previous objective.

Factor: x 2 + 12 x y + 36 y 2 .

Solution

x 2 + 12 x y + 36 y 2 Note that the first terms are x , last terms contain y . ( x _ y ) ( x _ y )

Find the numbers that multiply to 36 and add to 12.

Factors of 36 Sum of factors
1, 36 1 + 36 = 37
2, 18 2 + 18 = 20
3, 12 3 + 12 = 15
4, 9 4 + 9 = 13
6, 6 6 + 6 = 12 *

Use 6 and 6 as the coefficients of the last terms. ( x + 6 y ) ( x + 6 y ) Check your answer. ( x + 6 y ) ( x + 6 y ) x 2 + 6 x y + 6 x y + 36 y 2 x 2 + 12 x y + 36 y 2

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Factor: u 2 + 11 u v + 28 v 2 .

( u + 4 v ) ( u + 7 v )

Got questions? Get instant answers now!

Factor: x 2 + 13 x y + 42 y 2 .

( x + 6 y ) ( x + 7 y )

Got questions? Get instant answers now!

Factor: r 2 8 r x 9 s 2 .

Solution

We need r in the first term of each binomial and s in the second term. The last term of the trinomial is negative, so the factors must have opposite signs.

r 2 8 r x 9 s 2 Note that the first terms are r , last terms contain s . ( r _ s ) ( r _ s )

Find the numbers that multiply to −9 and add to −8 .

Factors of −9 Sum of factors
1 , −9 1 + ( −9 ) = −8 *
−1 , 9 −1 + 9 = 8
3 , −3 3 + ( −3 ) = 0

Use 1 , −9 as coefficients of the last terms. ( r + s ) ( r 9 s ) Check your answer. ( r 9 s ) ( r + s ) r 2 + r s 9 r s 9 s 2 r 2 8 r s 9 s 2

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Factor: a 2 11 a b + 10 b 2 .

( a b ) ( a 10 b )

Got questions? Get instant answers now!

Factor: m 2 13 m n + 12 n 2 .

( m n ) ( m 12 n )

Got questions? Get instant answers now!

Factor: u 2 9 u v 12 v 2 .

Solution

We need u in the first term of each binomial and v in the second term. The last term of the trinomial is negative, so the factors must have opposite signs.

u 2 9 u v 12 v 2 Note that the first terms are u , last terms contain v . ( u _ v ) ( u _ v )

Find the numbers that multiply to −12 and add to −9 .

Factors of −12 Sum of factors
1 , −12 1 + ( −12 ) = −11
−1 , 12 −1 + 12 = 11
2 , −6 2 + ( −6 ) = −4
−2 , 6 −2 + 6 = 4
3 , −4 3 + ( −4 ) = −1
−3 , 4 −3 + 4 = 1

Note there are no factor pairs that give us −9 as a sum. The trinomial is prime.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Factor: x 2 7 x y 10 y 2 .

prime

Got questions? Get instant answers now!

Factor: p 2 + 15 p q + 20 q 2 .

prime

Got questions? Get instant answers now!

Key concepts

  • Factor trinomials of the form x 2 + b x + c
    1. Write the factors as two binomials with first terms x : ( x ) ( x ) .
    2. Find two numbers m and n that
      Multiply to c , m · n = c
      Add to b , m + n = b
    3. Use m and n as the last terms of the factors: ( x + m ) ( x + n ) .
    4. Check by multiplying the factors.

Practice makes perfect

Factor Trinomials of the Form x 2 + b x + c

In the following exercises, factor each trinomial of the form x 2 + b x + c .

x 2 + 4 x + 3

( x + 1 ) ( x + 3 )

Got questions? Get instant answers now!

m 2 + 12 m + 11

( m + 1 ) ( m + 11 )

Got questions? Get instant answers now!

a 2 + 9 a + 20

( a + 4 ) ( a + 5 )

Got questions? Get instant answers now!

p 2 + 11 p + 30

( p + 5 ) ( p + 6 )

Got questions? Get instant answers now!

n 2 + 19 n + 48

( n + 3 ) ( n + 16 )

Got questions? Get instant answers now!

a 2 + 25 a + 100

( a + 5 ) ( a + 20 )

Got questions? Get instant answers now!

x 2 8 x + 12

( x 2 ) ( x 6 )

Got questions? Get instant answers now!

y 2 18 x + 45

( y 3 ) ( y 15 )

Got questions? Get instant answers now!

x 2 8 x + 7

( x 1 ) ( x 7 )

Got questions? Get instant answers now!

p 2 + 5 p 6

( p 1 ) ( p + 6 )

Got questions? Get instant answers now!

y 2 6 y 7

( y + 1 ) ( y 7 )

Got questions? Get instant answers now!

x 2 x 12

( x 4 ) ( x + 1 ) ( x 4 ) ( x + 3 )

Got questions? Get instant answers now!

a 2 3 a 28

( a 7 ) ( a + 4 )

Got questions? Get instant answers now!

w 2 5 w 36

( w 9 ) ( w + 4 )

Got questions? Get instant answers now!

8 6 x + x 2

( x 4 ) ( x 2 )

Got questions? Get instant answers now!

x 2 12 11 x

( x 12 ) ( x + 1 )

Got questions? Get instant answers now!

Factor Trinomials of the Form x 2 + b x y + c y 2

In the following exercises, factor each trinomial of the form x 2 + b x y + c y 2 .

p 2 + 3 p q + 2 q 2

( p + q ) ( p + 2 q )

Got questions? Get instant answers now!

r 2 + 15 r s + 36 s 2

( r + 3 s ) ( r + 12 s )

Got questions? Get instant answers now!

m 2 12 m n + 20 n 2

( m 2 n ) ( m 10 n )

Got questions? Get instant answers now!

x 2 2 x y 80 y 2

( x + 8 y ) ( x 10 y )

Got questions? Get instant answers now!

p 2 8 p q 65 q 2

Got questions? Get instant answers now!

m 2 64 m n 65 n 2

( m + n ) ( m 65 n )

Got questions? Get instant answers now!

p 2 2 p q 35 q 2

Got questions? Get instant answers now!

a 2 + 5 a b 24 b 2

( a + 8 b ) ( a 3 b )

Got questions? Get instant answers now!

x 2 3 x y 14 y 2

prime

Got questions? Get instant answers now!

u 2 8 u v 24 v 2

Got questions? Get instant answers now!

m 2 5 m n + 30 n 2

prime

Got questions? Get instant answers now!

Mixed Practice

In the following exercises, factor each expression.

u 2 12 u + 36

( u 6 ) ( u 6 )

Got questions? Get instant answers now!

x 2 14 x 32

( x + 2 ) ( x 16 )

Got questions? Get instant answers now!

r 2 20 r s + 64 s 2

( r 4 s ) ( r 16 s )

Got questions? Get instant answers now!

k 2 + 34 k + 120

( k + 4 ) ( k + 30 )

Got questions? Get instant answers now!

m 2 + m n 56 n 2

( m + 8 n ) ( m 7 n )

Got questions? Get instant answers now!

q 2 29 q r 96 r 2

Got questions? Get instant answers now!

u 2 17 u v + 30 v 2

( u 15 v ) ( u 2 v )

Got questions? Get instant answers now!

c 2 8 c d + 26 d 2

prime

Got questions? Get instant answers now!

Everyday math

Consecutive integers Deirdre is thinking of two consecutive integers whose product is 56. The trinomial x 2 + x 56 describes how these numbers are related. Factor the trinomial.

( x + 8 ) ( x 7 )

Got questions? Get instant answers now!

Consecutive integers Deshawn is thinking of two consecutive integers whose product is 182. The trinomial x 2 + x 182 describes how these numbers are related. Factor the trinomial.

Got questions? Get instant answers now!

Writing exercises

Many trinomials of the form x 2 + b x + c factor into the product of two binomials ( x + m ) ( x + n ) . Explain how you find the values of m and n .

Answers may vary

Got questions? Get instant answers now!

How do you determine whether to use plus or minus signs in the binomial factors of a trinomial of the form x 2 + b x + c where b and c may be positive or negative numbers?

Got questions? Get instant answers now!

Will factored x 2 x 20 as ( x + 5 ) ( x 4 ) . Bill factored it as ( x + 4 ) ( x 5 ) . Phil factored it as ( x 5 ) ( x 4 ) . Who is correct? Explain why the other two are wrong.

Answers may vary

Got questions? Get instant answers now!

Look at [link] , where we factored y 2 + 17 y + 60 . We made a table listing all pairs of factors of 60 and their sums. Do you find this kind of table helpful? Why or why not?

Got questions? Get instant answers now!

Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has the following statements all to be preceded by “I can…”. The first is “factor trinomials of the form x ^ 2 +b x + c”. The second is “factor trinomials of the form x^2 + b x y + c y ^ 2”. In the columns beside these statements are the headers, “confidently”, “with some help”, and “no-I don’t get it!”.

After reviewing this checklist, what will you do to become confident for all goals?

Questions & Answers

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
what is antagonism?
mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
Linsy
how many types of solid did we have
Samuel Reply
what is an ionic bond
Samuel
What is Atoms
Daprince Reply
what is fallopian tube
Merolyn
what is bladder
Merolyn
what's bulbourethral gland
Eduek Reply
urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
what is heart
Konadu Reply
how is urine formed in human
Konadu
how is urine formed in human
Rahma
what is the diference between a cavity and a canal
Pelagie Reply
what is the causative agent of malaria
Diamond
malaria is caused by an insect called mosquito.
Naomi
Malaria is cause by female anopheles mosquito
Isaac
Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
Olalekan
a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
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. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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