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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

Ayele, K., 2003. Introductory Economics, 3rd ed., Addis Ababa.
Widad Reply
can you send the book attached ?
Ariel
?
Ariel
What is economics
Widad Reply
the study of how humans make choices under conditions of scarcity
AI-Robot
U(x,y) = (x×y)1/2 find mu of x for y
Desalegn Reply
U(x,y) = (x×y)1/2 find mu of x for y
Desalegn
what is ecnomics
Jan Reply
this is the study of how the society manages it's scarce resources
Belonwu
what is macroeconomic
John Reply
macroeconomic is the branch of economics which studies actions, scale, activities and behaviour of the aggregate economy as a whole.
husaini
etc
husaini
difference between firm and industry
husaini Reply
what's the difference between a firm and an industry
Abdul
firm is the unit which transform inputs to output where as industry contain combination of firms with similar production 😅😅
Abdulraufu
Suppose the demand function that a firm faces shifted from Qd  120 3P to Qd  90  3P and the supply function has shifted from QS  20  2P to QS 10  2P . a) Find the effect of this change on price and quantity. b) Which of the changes in demand and supply is higher?
Toofiq Reply
explain standard reason why economic is a science
innocent Reply
factors influencing supply
Petrus Reply
what is economic.
Milan Reply
scares means__________________ends resources. unlimited
Jan
economics is a science that studies human behaviour as a relationship b/w ends and scares means which have alternative uses
Jan
calculate the profit maximizing for demand and supply
Zarshad Reply
Why qualify 28 supplies
Milan
what are explicit costs
Nomsa Reply
out-of-pocket costs for a firm, for example, payments for wages and salaries, rent, or materials
AI-Robot
concepts of supply in microeconomics
David Reply
economic overview notes
Amahle Reply
identify a demand and a supply curve
Salome Reply
i don't know
Parul
there's a difference
Aryan
Demand curve shows that how supply and others conditions affect on demand of a particular thing and what percent demand increase whith increase of supply of goods
Israr
Hi Sir please how do u calculate Cross elastic demand and income elastic demand?
Abari
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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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