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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 .

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

Sample set e

3 x .

The coefficient of x is 3.

6 a 3 .

The coefficient of a 3 is 6.

9 ( 4 a ) .

The coefficient of ( 4 a ) is 9.

3 8 x y 4 .

The coefficient of x y 4 is 3 8 .

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 .

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 ) .

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

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

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

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.

Why is the number 14 considered to be an expression?

Why is the number x considered to be an expression?

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

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

2 x + 1

6 x 10

two : 6 x , 10

2 x 3 + x 15

5 x 2 + 6 x 2

three : 5 x 2 , 6 x , 2

3 x

5 c z

one : 5 c z

2

61

one : 61

x

4 y 3

one : 4 y 3

17 a b 2

a + 1

two : a , 1

( a + 1 )

2 x + x + 7

three : 2 x , x , 7

2 x + ( x + 7 )

( a + 1 ) + ( a 1 )

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

a + 1 + ( a 1 )

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

11 y 3 33 y 3

45 a b 2 + 9 b 2

9 b 2

6 x 2 y 3 + 18 x 2

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

( a + b )

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

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

2 a b 2 c 2

4 x 2 y + 5 a 2 b

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

( a 3 )

15 x 2 30 x y 2

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

no commom factors

0.06 a b 2 + 0.03 a

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

( a + 7 )

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

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

9 32

For the following problems, note how many:

a ' s in 4 a ?

z ' s in 12 z ?

12

x 2 's in 5 x 2 ?

y 3 's in 6 y 3 ?

6

x y ' s in 9 x y ?

a 2 b ' s in 10 a 2 b ?

10

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

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

8

y 2 's in 3 x 3 y 2 ?

12 x ' s in 12 x 2 y 5 ?

x y 5

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

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

5 x

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

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

x 2 ( x 7 )

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

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.

7 y ; y

7

10 x ; x

5 a ; 5

a

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

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

6 x 2 b 2

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

9 a 2 b 5 ; 3 a b 3

3 a b 2

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

( 4 ) a 6 b 2 ; a b

( 4 ) a 5 b

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

Exercises for review

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

16 x 16 ( x 1 ) 12

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

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

50

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

( [link] ) Express 0.0000152 using scientific notation.

1.52 × 10 5

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Source:  OpenStax, Basic mathematics review. OpenStax CNX. Jun 06, 2012 Download for free at http://cnx.org/content/col11427/1.2
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