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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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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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