<< Chapter < Page Chapter >> Page >

( a + b ) ( c + d ) = a c + a d + b c + b d

This method is commonly called the FOIL method .

  • F First terms
  • O Outer terms
  • I Inner terms
  • L Last terms

( a + b ) ( 2 + 3 ) = ( a + b ) + ( a + b ) 2 terms + ( a + b ) + ( a + b ) + ( a + b ) 3 terms

Rearranging,

= a + a + b + b + a + a + a + b + b + b = 2 a + 2 b + 3 a + 3 b

Combining like terms,

= 5 a + 5 b

This use of the distributive property suggests the following rule.

Multiplying a polynomial by a polynomial

To multiply two polynomials together, multiply every term of one polynomial by every term of the other polynomial.

Sample set c

Perform the following multiplications and simplify.

Finding the product of 'a plus six' and 'a plus three' using the FOIL method. See the longdesc for a full description.

With some practice, the second and third terms can be combined mentally.

Finding the product of two binomials 'x plus y' and 'two x plus four y' using the FOIL method. See the longdesc for a full description.

Finding the product of two polynomials 'x squared plus four' and 'x squared plus seven x plus two' using the FOIL method. See the longdesc for a full description.

Finding the product of two binomials 'a minus four' and 'a minus three' using the FOIL method. See the longdesc for a full description.

( m 3 ) 2 = ( m 3 ) ( m 3 ) = m m + m ( 3 ) 3 m 3 ( 3 ) = m 2 3 m 3 m + 9 = m 2 6 m + 9

( x + 5 ) 3 = ( x + 5 ) ( x + 5 ) ( x + 5 ) Associate the first two factors . = [ ( x + 5 ) ( x + 5 ) ] ( x + 5 ) = [ x 2 + 5 x + 5 x + 25 ] ( x + 5 ) = [ x 2 + 10 x + 25 ] ( x + 5 ) = x 2 x + x 2 5 + 10 x x + 10 x 5 + 25 x + 25 5 = x 3 + 5 x 2 + 10 x 2 + 50 x + 25 x + 125 = x 3 + 15 x 2 + 75 x + 125

Practice set c

Find the following products and simplify.

( a + 1 ) ( a + 4 )

a 2 + 5 a + 4

( m 9 ) ( m 2 )

m 2 11 m + 18

( 2 x + 4 ) ( x + 5 )

2 x 2 + 14 x + 20

( x + y ) ( 2 x 3 y )

2 x 2 x y 3 y 2

( 3 a 2 1 ) ( 5 a 2 + a )

15 a 4 + 3 a 3 5 a 2 a

( 2 x 2 y 3 + x y 2 ) ( 5 x 3 y 2 + x 2 y )

10 x 5 y + 5 7 x 4 y 4 + x 3 y 3

( a + 3 ) ( a 2 + 3 a + 6 )

a 3 + 6 a 2 + 15 a + 18

( a + 4 ) ( a + 4 )

a 2 + 8 a + 16

( r 7 ) ( r 7 )

r 2 14 r + 49

( x + 6 ) 2

x 2 + 12 x + 36

( y 8 ) 2

y 2 16 y + 64

Sample set d

Perform the following additions and subtractions.

3 x + 7 + ( x 3 ) . We must first remove the parentheses . They are preceded by a " + " sign, so we remove them and leave the sign of each term the same . 3 x + 7 + x 3 Combine like terms . 4 x + 4

5 y 3 + 11 ( 12 y 3 2 ) . We first remove the parentheses . They are preceded by a "-" sign, so we remove them and change the sign of each term inside them . 5 y 3 + 11 12 y 3 + 2 Combine like terms . 7 y 3 + 13

Add 4 x 2 + 2 x 8 to 3 x 2 7 x 10 .

( 4 x 2 + 2 x 8 ) + ( 3 x 2 7 x 10 ) 4 x 2 + 2 x 8 + 3 x 2 7 x 10 7 x 2 5 x 18

Subtract 8 x 2 5 x + 2 from 3 x 2 + x 12 .

( 3 x 2 + x 12 ) ( 8 x 2 5 x + 2 ) 3 x 2 + x 12 8 x 2 + 5 x 2 5 x 2 + 6 x 14

Be very careful not to write this problem as

3 x 2 + x 12 8 x 2 5 x + 2

This form has us subtracting only the very first term, 8 x 2 , rather than the entire expression. Use parentheses.
Another incorrect form is

8 x 2 5 x + 2 ( 3 x 2 + x 12 )

This form has us performing the subtraction in the wrong order.

Practice set d

Perform the following additions and subtractions.

6 y 2 + 2 y 1 + ( 5 y 2 18 )

11 y 2 + 2 y 19

( 9 m n ) ( 10 m + 12 n )

m 13 n

Add 2 r 2 + 4 r 1 to 3 r 2 r 7 .

5 r 2 + 3 r 8

Subtract 4 s 3 from 7 s + 8 .

3 s + 11

Exercises

For the following problems, perform the multiplications and combine any like terms.

7 ( x + 6 )

7 x + 42

4 ( y + 3 )

6 ( y + 4 )

6 y + 24

8 ( m + 7 )

5 ( a 6 )

5 a 30

2 ( x 10 )

3 ( 4 x + 2 )

12 x + 6

6 ( 3 x + 4 )

9 ( 4 y 3 )

36 y 27

5 ( 8 m 6 )

9 ( a + 7 )

9 a 63

3 ( b + 8 )

4 ( x + 2 )

4 x 8

6 ( y + 7 )

3 ( a 6 )

3 a + 18

9 ( k 7 )

5 ( 2 a + 1 )

10 a 5

7 ( 4 x + 2 )

3 ( 10 y 6 )

30 y + 18

8 ( 4 y 11 )

x ( x + 6 )

x 2 + 6 x

y ( y + 7 )

m ( m 4 )

m 2 4 m

k ( k 11 )

3 x ( x + 2 )

3 x 2 + 6 x

4 y ( y + 7 )

6 a ( a 5 )

6 a 2 30 a

9 x ( x 3 )

3 x ( 5 x + 4 )

15 x 2 + 12 x

4 m ( 2 m + 7 )

2 b ( b 1 )

2 b 2 2 b

7 a ( a 4 )

3 x 2 ( 5 x 2 + 4 )

15 x 4 + 12 x 2

9 y 3 ( 3 y 2 + 2 )

4 a 4 ( 5 a 3 + 3 a 2 + 2 a )

20 a 7 + 12 a 6 + 8 a 5

2 x 4 ( 6 x 3 5 x 2 2 x + 3 )

5 x 2 ( x + 2 )

5 x 3 10 x 2

6 y 3 ( y + 5 )

2 x 2 y ( 3 x 2 y 2 6 x )

6 x 4 y 3 12 x 3 y

8 a 3 b 2 c ( 2 a b 3 + 3 b )

b 5 x 2 ( 2 b x 11 )

2 b 6 x 3 11 b 5 x 2

4 x ( 3 x 2 6 x + 10 )

9 y 3 ( 2 y 4 3 y 3 + 8 y 2 + y 6 )

18 y 7 27 y 6 + 72 y 5 + 9 y 4 54 y 3

a 2 b 3 ( 6 a b 4 + 5 a b 3 8 b 2 + 7 b 2 )

( a + 4 ) ( a + 2 )

a 2 + 6 a + 8

( x + 1 ) ( x + 7 )

( y + 6 ) ( y 3 )

y 2 + 3 y 18

( t + 8 ) ( t 2 )

( i 3 ) ( i + 5 )

i 2 + 2 i 15

( x y ) ( 2 x + y )

( 3 a 1 ) ( 2 a 6 )

6 a 2 20 a + 6

( 5 a 2 ) ( 6 a 8 )

( 6 y + 11 ) ( 3 y + 10 )

18 y 2 + 93 y + 110

( 2 t + 6 ) ( 3 t + 4 )

( 4 + x ) ( 3 x )

x 2 x + 12

( 6 + a ) ( 4 + a )

( x 2 + 2 ) ( x + 1 )

x 3 + x 2 + 2 x + 2

( x 2 + 5 ) ( x + 4 )

( 3 x 2 5 ) ( 2 x 2 + 1 )

6 x 4 7 x 2 5

( 4 a 2 b 3 2 a ) ( 5 a 2 b 3 b )

( 6 x 3 y 4 + 6 x ) ( 2 x 2 y 3 + 5 y )

12 x 5 y 7 + 30 x 3 y 5 + 12 x 3 y 3 + 30 x y

5 ( x 7 ) ( x 3 )

4 ( a + 1 ) ( a 8 )

4 a 2 28 a 32

a ( a 3 ) ( a + 5 )

x ( x + 1 ) ( x + 4 )

x 3 + 5 x 2 + 4 x

x 2 ( x + 5 ) ( x + 7 )

y 3 ( y 3 ) ( y 2 )

y 5 5 y 4 + 6 y 3

2 a 2 ( a + 4 ) ( a + 3 )

5 y 6 ( y + 7 ) ( y + 1 )

5 y 8 + 40 y 7 + 35 y 6

a b 2 ( a 2 2 b ) ( a + b 4 )

x 3 y 2 ( 5 x 2 y 2 3 ) ( 2 x y 1 )

10 x 6 y 5 5 x 5 y 4 6 x 4 y 3 + 3 x 3 y 2

6 ( a 2 + 5 a + 3 )

8 ( c 3 + 5 c + 11 )

8 c 3 + 40 c + 88

3 a 2 ( 2 a 3 10 a 2 4 a + 9 )

6 a 3 b 3 ( 4 a 2 b 6 + 7 a b 8 + 2 b 10 + 14 )

24 a 5 b 9 + 42 a 4 b 11 + 12 a 3 b 13 + 18 a 3 b 3

( a 4 ) ( a 2 + a 5 )

( x 7 ) ( x 2 + x 3 )

x 3 6 x 2 10 x + 21

( 2 x + 1 ) ( 5 x 3 + 6 x 2 + 8 )

( 7 a 2 + 2 ) ( 3 a 5 4 a 3 a 1 )

21 a 7 22 a 5 15 a 3 7 a 2 2 a 2

( x + y ) ( 2 x 2 + 3 x y + 5 y 2 )

( 2 a + b ) ( 5 a 2 + 4 a 2 b b 4 )

10 a 3 + 8 a 3 b + 4 a 2 b 2 + 5 a 2 b b 2 8 a 4 b 2 a b

( x + 3 ) 2

( x + 1 ) 2

x 2 + 2 x + 1

( x 5 ) 2

( a + 2 ) 2

a 2 + 4 a + 4

( a 9 ) 2

( 3 x 5 ) 2

9 x 2 + 30 x 25

( 8 t + 7 ) 2

For the following problems, perform the indicated operations and combine like terms.

3 x 2 + 5 x 2 + ( 4 x 2 10 x 5 )

7 x 2 5 x 7

2 x 3 + 4 x 2 + 5 x 8 + ( x 3 3 x 2 11 x + 1 )

5 x 12 x y + 4 y 2 + ( 7 x + 7 x y 2 y 2 )

2 y 2 5 x y 12 x

( 6 a 2 3 a + 7 ) 4 a 2 + 2 a 8

( 5 x 2 24 x 15 ) + x 2 9 x + 14

6 x 2 33 x 1

( 3 x 3 7 x 2 + 2 ) + ( x 3 + 6 )

( 9 a 2 b 3 a b + 12 a b 2 ) + a b 2 + 2 a b

9 a 2 b + 13 a b 2 a b

6 x 2 12 x + ( 4 x 2 3 x 1 ) + 4 x 2 10 x 4

5 a 3 2 a 26 + ( 4 a 3 11 a 2 + 2 a ) 7 a + 8 a 3 + 20

17 a 3 11 a 2 7 a 6

2 x y 15 ( 5 x y + 4 )

Add 4 x + 6 to 8 x 15 .

12 x 9

Add 5 y 2 5 y + 1 to 9 y 2 + 4 y 2 .

Add 3 ( x + 6 ) to 4 ( x 7 ) .

7 x 10

Add 2 ( x 2 4 ) to 5 ( x 2 + 3 x 1 ) .

Add four times 5 x + 2 to three times 2 x 1 .

26 x + 5

Add five times 3 x + 2 to seven times 4 x + 3 .

Add 4 times 9 x + 6 to 2 times 8 x 3 .

20 x 18

Subtract 6 x 2 10 x + 4 from 3 x 2 2 x + 5 .

Substract a 2 16 from a 2 16 .

0

Exercises for review

( [link] ) Simplify ( 15 x 2 y 6 5 x y 2 ) 4 .

( [link] ) Express the number 198,000 using scientific notation.

1.98 × 10 5

( [link] ) How many 4 a 2 x 3 ' s are there in 16 a 4 x 5 ?

( [link] ) State the degree of the polynomial 4 x y 3 + 3 x 5 y 5 x 3 y 3 , and write the numerical coefficient of each term.

degree is 6 ; 4 , 3 , 5

( [link] ) Simplify 3 ( 4 x 5 ) + 2 ( 5 x 2 ) ( x 3 ) .

Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra i for the community college. OpenStax CNX. Dec 19, 2014 Download for free at http://legacy.cnx.org/content/col11598/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra i for the community college' conversation and receive update notifications?

Ask