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5 x 3 ( y 7 ) means there are five x 3 ( y 7 ) ' s . It could also mean there are 5 x 3 ( x 7 ) ' s . It could also mean there are 5 ( x 7 ) x 3 ' s .

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Practice set d

What does the coefficient of a quantity tell us?

The difference between coefficients and exponents

It is important to keep in mind the difference between coefficients and exponents .

Coefficients record the number of like terms in an algebraic expression.
x + x + x + x 4 terms = 4 x coefficient is 4
Exponents record the number of like factors in a term.
x x x x 4 factors = x 4 exponent is 4

In a term, the coefficient of a particular group of factors is the remaining group of factors.

how many of that quantity there are

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Sample set e

3 x .

The coefficient of x is 3.

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6 a 3 .

The coefficient of a 3 is 6.

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9 ( 4 a ) .

The coefficient of ( 4 a ) is 9.

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3 8 x y 4 .

The coefficient of x y 4 is 3 8 .

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3 x 2 y .

The coefficient of x 2 y is 3; the coefficient of y is 3 x 2 ; and the coefficient of 3 is x 2 y .

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4 ( x + y ) 2 .

The coefficient of ( x + y ) 2 is 4; the coefficient of 4 is ( x + y ) 2 ; and the coefficient of ( x + y ) is 4 ( x + y ) since 4 ( x + y ) 2 can be written as 4 ( x + y ) ( x + y ) .

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Practice set e

Determine the coefficients.
In the term 6 x 3 , the coefficient of
(a) x 3 is .
(b) 6 is .

(a) 6 (b) x 3

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In the term 3 x ( y 1 ) , the coefficient of
(a) x ( y 1 ) is .
(b) ( y 1 ) is .
(c) 3 ( y 1 ) is .
(d) x is .
(e) 3 is .
(f) The numerical coefficient is .

(a) 3 (b) 3 x (c) x (d) 3 ( y 1 ) (e) x ( y 1 ) (f) 3

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In the term 10 a b 4 , the coefficient of
(a) a b 4 is .
(b) b 4 is .
(c) a is .
(d) 10 is .
(e) 10 a b 3 is .

(a) 10 (b) 10 a (c) 10 b 4 (d) a b 4 (e) b

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Exercises

What is an algebraic expression?

An algebraic expression is a number, a letter, or a collection of numbers and letters along with meaningful signs of operation.

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Why is the number 14 considered to be an expression?

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Why is the number x considered to be an expression?

x is an expression because it is a letter (see the definition).

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For the expressions in the following problems, write the number of terms that appear and then list the terms.

6 x 10

two : 6 x , 10

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5 x 2 + 6 x 2

three : 5 x 2 , 6 x , 2

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5 c z

one : 5 c z

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4 y 3

one : 4 y 3

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a + 1

two : a , 1

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2 x + x + 7

three : 2 x , x , 7

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( a + 1 ) + ( a 1 )

two : ( a + b ) , ( a 1 )

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For the following problems, list, if any should appear, the common factors in the expressions.

x 2 + 5 x 2 2 x 2

x 2

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2 ( a + b ) 3 ( a + b )

( a + b )

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8 a 2 ( b + 1 ) 10 a 2 ( b + 1 )

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14 a b 2 c 2 ( c + 8 ) + 12 a b 2 c 2

2 a b 2 c 2

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

( a 3 )

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12 a 3 b 2 c 7 ( b + 1 ) ( c a )

no commom factors

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5.2 ( a + 7 ) 2 + 17.1 ( a + 7 )

( a + 7 )

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3 4 x 2 y 2 z 2 + 3 8 x 2 z 2

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9 16 ( a 2 b 2 ) + 9 32 ( b 2 a 2 )

9 32

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For the following problems, note how many:

z ' s in 12 z ?

12

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x 2 's in 5 x 2 ?

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y 3 's in 6 y 3 ?

6

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x y ' s in 9 x y ?

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a 2 b ' s in 10 a 2 b ?

10

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( a + 1 ) ' s in 4 ( a + 1 ) ?

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( 9 + y ) ' s in 8 ( 9 + y ) ?

8

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y 2 's in 3 x 3 y 2 ?

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12 x ' s in 12 x 2 y 5 ?

x y 5

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( a + 5 ) ' s in 2 ( a + 5 ) ?

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( x y ) ' s in 5 x ( x y ) ?

5 x

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( x + 1 ) ' s in 8 ( x + 1 ) ?

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2 ' s in 2 x 2 ( x 7 ) ?

x 2 ( x 7 )

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3 ( a + 8 ) ' s in 6 x 2 ( a + 8 ) 3 ( a 8 ) ?

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For the following problems, a term will be given followed by a group of its factors. List the coefficient of the given group of factors.

12 a 2 b 3 c 2 r 7 ; a 2 c 2 r 7

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6 x 2 b 2 ( c 1 ) ; c 1

6 x 2 b 2

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10 x ( x + 7 ) 2 ; 10 ( x + 7 )

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9 a 2 b 5 ; 3 a b 3

3 a b 2

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15 x 4 y 4 ( z + 9 a ) 3 ; ( z + 9 a )

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( 4 ) a 6 b 2 ; a b

( 4 ) a 5 b

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( 11 a ) ( a + 8 ) 3 ( a 1 ) ; ( a + 8 ) 2

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Exercises for review

( [link] ) Simplify [ 2 x 8 ( x 1 ) 5 x 4 ( x 1 ) 2 ] 4 .

16 x 16 ( x 1 ) 12

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( [link] ) Supply the missing phrase. Absolute value speaks to the question of and not "which way."

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( [link] ) Find the value of [ 6 ( 4 2 ) + 7 ( 3 + 5 ) ] .

50

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( [link] ) Find the value of 2 5 4 2 3 2 .

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( [link] ) Express 0.0000152 using scientific notation.

1.52 × 10 5

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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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