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( 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

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( 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

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Practice set c

Find the following products and simplify.

( a + 1 ) ( a + 4 )

a 2 + 5 a + 4

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( m 9 ) ( m 2 )

m 2 11 m + 18

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( 2 x + 4 ) ( x + 5 )

2 x 2 + 14 x + 20

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( x + y ) ( 2 x 3 y )

2 x 2 x y 3 y 2

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

15 a 4 + 3 a 3 5 a 2 a

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( 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

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

a 3 + 6 a 2 + 15 a + 18

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( a + 4 ) ( a + 4 )

a 2 + 8 a + 16

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( r 7 ) ( r 7 )

r 2 14 r + 49

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( x + 6 ) 2

x 2 + 12 x + 36

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( y 8 ) 2

y 2 16 y + 64

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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

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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

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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

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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.

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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

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

m 13 n

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

5 r 2 + 3 r 8

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Subtract 4 s 3 from 7 s + 8 .

3 s + 11

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Exercises

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

5 ( a 6 )

5 a 30

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9 ( 4 y 3 )

36 y 27

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9 ( a + 7 )

9 a 63

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4 ( x + 2 )

4 x 8

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3 ( a 6 )

3 a + 18

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5 ( 2 a + 1 )

10 a 5

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3 ( 10 y 6 )

30 y + 18

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m ( m 4 )

m 2 4 m

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3 x ( x + 2 )

3 x 2 + 6 x

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6 a ( a 5 )

6 a 2 30 a

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3 x ( 5 x + 4 )

15 x 2 + 12 x

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2 b ( b 1 )

2 b 2 2 b

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3 x 2 ( 5 x 2 + 4 )

15 x 4 + 12 x 2

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

20 a 7 + 12 a 6 + 8 a 5

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

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5 x 2 ( x + 2 )

5 x 3 10 x 2

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

6 x 4 y 3 12 x 3 y

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

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b 5 x 2 ( 2 b x 11 )

2 b 6 x 3 11 b 5 x 2

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4 x ( 3 x 2 6 x + 10 )

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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

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

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( a + 4 ) ( a + 2 )

a 2 + 6 a + 8

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( y + 6 ) ( y 3 )

y 2 + 3 y 18

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( i 3 ) ( i + 5 )

i 2 + 2 i 15

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( 3 a 1 ) ( 2 a 6 )

6 a 2 20 a + 6

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( 5 a 2 ) ( 6 a 8 )

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

18 y 2 + 93 y + 110

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( 4 + x ) ( 3 x )

x 2 x + 12

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( x 2 + 2 ) ( x + 1 )

x 3 + x 2 + 2 x + 2

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

6 x 4 7 x 2 5

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

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( 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

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5 ( x 7 ) ( x 3 )

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4 ( a + 1 ) ( a 8 )

4 a 2 28 a 32

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x ( x + 1 ) ( x + 4 )

x 3 + 5 x 2 + 4 x

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y 3 ( y 3 ) ( y 2 )

y 5 5 y 4 + 6 y 3

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

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

5 y 8 + 40 y 7 + 35 y 6

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

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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

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8 ( c 3 + 5 c + 11 )

8 c 3 + 40 c + 88

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

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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

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( a 4 ) ( a 2 + a 5 )

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( x 7 ) ( x 2 + x 3 )

x 3 6 x 2 10 x + 21

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

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( 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

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

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( 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

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( x + 1 ) 2

x 2 + 2 x + 1

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( a + 2 ) 2

a 2 + 4 a + 4

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( 3 x 5 ) 2

9 x 2 + 30 x 25

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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

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

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

2 y 2 5 x y 12 x

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

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

6 x 2 33 x 1

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

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( 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

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

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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

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2 x y 15 ( 5 x y + 4 )

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Add 4 x + 6 to 8 x 15 .

12 x 9

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

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

7 x 10

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

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

26 x + 5

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

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

20 x 18

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

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Substract a 2 16 from a 2 16 .

0

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Exercises for review

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

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

1.98 × 10 5

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

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( [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

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

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Questions & Answers

calculate molarity of NaOH solution when 25.0ml of NaOH titrated with 27.2ml of 0.2m H2SO4
Gasin Reply
what's Thermochemistry
rhoda Reply
the study of the heat energy which is associated with chemical reactions
Kaddija
How was CH4 and o2 was able to produce (Co2)and (H2o
Edafe Reply
explain please
Victory
First twenty elements with their valences
Martine Reply
what is chemistry
asue Reply
what is atom
asue
what is the best way to define periodic table for jamb
Damilola Reply
what is the change of matter from one state to another
Elijah Reply
what is isolation of organic compounds
IKyernum Reply
what is atomic radius
ThankGod Reply
Read Chapter 6, section 5
Dr
Read Chapter 6, section 5
Kareem
Atomic radius is the radius of the atom and is also called the orbital radius
Kareem
atomic radius is the distance between the nucleus of an atom and its valence shell
Amos
Read Chapter 6, section 5
paulino
Bohr's model of the theory atom
Ayom Reply
is there a question?
Dr
when a gas is compressed why it becomes hot?
ATOMIC
It has no oxygen then
Goldyei
read the chapter on thermochemistry...the sections on "PV" work and the First Law of Thermodynamics should help..
Dr
Which element react with water
Mukthar Reply
Mgo
Ibeh
an increase in the pressure of a gas results in the decrease of its
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definition of the periodic table
Cosmos Reply
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what were atoms composed of?
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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