<< 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.

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

( 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

Got questions? Get instant answers now!

( 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

Got questions? Get instant answers now!

Practice set c

Find the following products and simplify.

( a + 1 ) ( a + 4 )

a 2 + 5 a + 4

Got questions? Get instant answers now!

( m 9 ) ( m 2 )

m 2 11 m + 18

Got questions? Get instant answers now!

( 2 x + 4 ) ( x + 5 )

2 x 2 + 14 x + 20

Got questions? Get instant answers now!

( x + y ) ( 2 x 3 y )

2 x 2 x y 3 y 2

Got questions? Get instant answers now!

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

15 a 4 + 3 a 3 5 a 2 a

Got questions? Get instant answers now!

( 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

Got questions? Get instant answers now!

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

a 3 + 6 a 2 + 15 a + 18

Got questions? Get instant answers now!

( a + 4 ) ( a + 4 )

a 2 + 8 a + 16

Got questions? Get instant answers now!

( r 7 ) ( r 7 )

r 2 14 r + 49

Got questions? Get instant answers now!

( x + 6 ) 2

x 2 + 12 x + 36

Got questions? Get instant answers now!

( y 8 ) 2

y 2 16 y + 64

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

m 13 n

Got questions? Get instant answers now!

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

5 r 2 + 3 r 8

Got questions? Get instant answers now!

Subtract 4 s 3 from 7 s + 8 .

3 s + 11

Got questions? Get instant answers now!

Exercises

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

5 ( a 6 )

5 a 30

Got questions? Get instant answers now!

9 ( 4 y 3 )

36 y 27

Got questions? Get instant answers now!

9 ( a + 7 )

9 a 63

Got questions? Get instant answers now!

4 ( x + 2 )

4 x 8

Got questions? Get instant answers now!

3 ( a 6 )

3 a + 18

Got questions? Get instant answers now!

5 ( 2 a + 1 )

10 a 5

Got questions? Get instant answers now!

3 ( 10 y 6 )

30 y + 18

Got questions? Get instant answers now!

m ( m 4 )

m 2 4 m

Got questions? Get instant answers now!

3 x ( x + 2 )

3 x 2 + 6 x

Got questions? Get instant answers now!

6 a ( a 5 )

6 a 2 30 a

Got questions? Get instant answers now!

3 x ( 5 x + 4 )

15 x 2 + 12 x

Got questions? Get instant answers now!

2 b ( b 1 )

2 b 2 2 b

Got questions? Get instant answers now!

3 x 2 ( 5 x 2 + 4 )

15 x 4 + 12 x 2

Got questions? Get instant answers now!

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

20 a 7 + 12 a 6 + 8 a 5

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

5 x 2 ( x + 2 )

5 x 3 10 x 2

Got questions? Get instant answers now!

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

6 x 4 y 3 12 x 3 y

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

b 5 x 2 ( 2 b x 11 )

2 b 6 x 3 11 b 5 x 2

Got questions? Get instant answers now!

4 x ( 3 x 2 6 x + 10 )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

( a + 4 ) ( a + 2 )

a 2 + 6 a + 8

Got questions? Get instant answers now!

( y + 6 ) ( y 3 )

y 2 + 3 y 18

Got questions? Get instant answers now!

( i 3 ) ( i + 5 )

i 2 + 2 i 15

Got questions? Get instant answers now!

( 3 a 1 ) ( 2 a 6 )

6 a 2 20 a + 6

Got questions? Get instant answers now!

( 5 a 2 ) ( 6 a 8 )

Got questions? Get instant answers now!

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

18 y 2 + 93 y + 110

Got questions? Get instant answers now!

( 4 + x ) ( 3 x )

x 2 x + 12

Got questions? Get instant answers now!

( x 2 + 2 ) ( x + 1 )

x 3 + x 2 + 2 x + 2

Got questions? Get instant answers now!

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

6 x 4 7 x 2 5

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

( 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

Got questions? Get instant answers now!

5 ( x 7 ) ( x 3 )

Got questions? Get instant answers now!

4 ( a + 1 ) ( a 8 )

4 a 2 28 a 32

Got questions? Get instant answers now!

x ( x + 1 ) ( x + 4 )

x 3 + 5 x 2 + 4 x

Got questions? Get instant answers now!

y 3 ( y 3 ) ( y 2 )

y 5 5 y 4 + 6 y 3

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

5 y 8 + 40 y 7 + 35 y 6

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

8 ( c 3 + 5 c + 11 )

8 c 3 + 40 c + 88

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

( a 4 ) ( a 2 + a 5 )

Got questions? Get instant answers now!

( x 7 ) ( x 2 + x 3 )

x 3 6 x 2 10 x + 21

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

( 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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

( 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

Got questions? Get instant answers now!

( x + 1 ) 2

x 2 + 2 x + 1

Got questions? Get instant answers now!

( a + 2 ) 2

a 2 + 4 a + 4

Got questions? Get instant answers now!

( 3 x 5 ) 2

9 x 2 + 30 x 25

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

2 y 2 5 x y 12 x

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

6 x 2 33 x 1

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

( 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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

2 x y 15 ( 5 x y + 4 )

Got questions? Get instant answers now!

Add 4 x + 6 to 8 x 15 .

12 x 9

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

7 x 10

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

26 x + 5

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

20 x 18

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Substract a 2 16 from a 2 16 .

0

Got questions? Get instant answers now!

Exercises for review

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

Got questions? Get instant answers now!

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

1.98 × 10 5

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

( [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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Questions & Answers

how do you get the 2/50
Abba Reply
number of sport play by 50 student construct discrete data
Aminu Reply
width of the frangebany leaves on how to write a introduction
Theresa Reply
Solve the mean of variance
Veronica Reply
Step 1: Find the mean. To find the mean, add up all the scores, then divide them by the number of scores. ... Step 2: Find each score's deviation from the mean. ... Step 3: Square each deviation from the mean. ... Step 4: Find the sum of squares. ... Step 5: Divide the sum of squares by n – 1 or N.
kenneth
what is error
Yakuba Reply
Is mistake done to something
Vutshila
Hy
anas
hy
What is the life teble
anas
hy
Jibrin
statistics is the analyzing of data
Tajudeen Reply
what is statics?
Zelalem Reply
how do you calculate mean
Gloria Reply
diveving the sum if all values
Shaynaynay
let A1,A2 and A3 events be independent,show that (A1)^c, (A2)^c and (A3)^c are independent?
Fisaye Reply
what is statistics
Akhisani Reply
data collected all over the world
Shaynaynay
construct a less than and more than table
Imad Reply
The sample of 16 students is taken. The average age in the sample was 22 years with astandard deviation of 6 years. Construct a 95% confidence interval for the age of the population.
Aschalew Reply
Bhartdarshan' is an internet-based travel agency wherein customer can see videos of the cities they plant to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400 a. what is the probability of getting more than 12,000 hits? b. what is the probability of getting fewer than 9,000 hits?
Akshay Reply
Bhartdarshan'is an internet-based travel agency wherein customer can see videos of the cities they plan to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400. a. What is the probability of getting more than 12,000 hits
Akshay
1
Bright
Sorry i want to learn more about this question
Bright
Someone help
Bright
a= 0.20233 b=0.3384
Sufiyan
a
Shaynaynay
How do I interpret level of significance?
Mohd Reply
It depends on your business problem or in Machine Learning you could use ROC- AUC cruve to decide the threshold value
Shivam
how skewness and kurtosis are used in statistics
Owen Reply
yes what is it
Taneeya
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