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MathMl with encoding

Module by: Herminder Singh. E-mail the author

Summary: MathMl with encoding

Overview

  • Algebraic Expressions
  • Terms and Factors
  • Common Factors
  • Coefficients

Algebraic Expressions

Algebraic Expression

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

Expressions

Algebraic expressions are often referred to simply as expressions, as in the following examples:

Example 1

x+4 x+4 is an expression.

Example 2

7y 7y is an expression.

Example 3

x-3 x 2 y 7+9x x-3 x 2 y 7+9x is an expression.

Example 4

The number 8 is an expression. 8 can be written with explicit signs of operation by writing it as 8+0 8+0 or 8·1 8·1 .

3 x 2 +6=4x-1 3 x 2 +6=4x-1 is not an expression, it is an equation. We will study equations in the next section.

Terms and Factors

Terms

In an algebraic expression, the quantities joined by "+" "+" signs are called terms.

In some expressions it will appear that terms are joined by "-" "-" signs. We must keep in mind that subtraction is addition of the negative, that is, a-b=a+(-b) a-b=a+(-b) .

An important concept that all students of algebra must be aware of is the difference between terms and factors.

Factors

Any numbers or symbols that are multiplied together are factors of their product.

Terms are parts of sums and are therefore joined by addition (or subtraction) signs.
Factors are parts of products and are therefore joined by multiplication signs.

Sample Set A

Identify the terms in the following expressions.

Example 5

3 x 4 +6 x 2 +5x+8 3 x 4 +6 x 2 +5x+8 .

This expression has four terms: 3 x 4 ,6 x 2 ,?5x, 3 x 4 ,6 x 2 ,?5x, and 8.

Example 6

15 y 8 15 y 8 .

In this expression there is only one term. The term is 15 y 8 15 y 8 .

Example 7

14 x 5 y+ (a+3) 2 14 x 5 y+ (a+3) 2 .

In this expression there are two terms: the terms are 14 x 5 y 14 x 5 y and (a+3) 2 (a+3) 2 . Notice that the term (a+3) 2 (a+3) 2 is itself composed of two like factors, each of which is composed of the two terms, a a and 3.

Example 8

m 3 -3 m 3 -3 .

Using our definition of subtraction, this expression can be written in the form m 3 +(-3) m 3 +(-3) . Now we can see that the terms are m 3 m 3 and -3 -3 .

Rather than rewriting the expression when a subtraction occurs, we can identify terms more quickly by associating the + + or - - sign with the individual quantity.

Example 9

p 4 -7 p 3 -2p-11 p 4 -7 p 3 -2p-11 .

Associating the sign with the individual quantities we see that the terms of this expression are p 4 ,?-7 p 3 ,?-2p, p 4 ,?-7 p 3 ,?-2p, and -11 -11 .

Practice Set A

Exercise 1

Let’s say it again. The difference between terms and factors is that terms are joined by
signs and factors are joined by
signs.

Solution

addition, multiplication

List the terms in the following expressions.

Exercise 2

4 x 2 -8x+7 4 x 2 -8x+7

Solution

4 x 2 ,?-8x,?7 4 x 2 ,?-8x,?7

Exercise 3

2xy+6 x 2 + (x-y) 4 2xy+6 x 2 + (x-y) 4

Solution

2xy,?6 x 2 ,? (x-y) 4 2xy,?6 x 2 ,? (x-y) 4

Exercise 4

5 x 2 +3x-3x y 7 +(x-y)( x 3 -6) 5 x 2 +3x-3x y 7 +(x-y)( x 3 -6)

Solution

5 x 2 ,3x,-3x y 7 ,?(x-y)( x 3 -6) 5 x 2 ,3x,-3x y 7 ,?(x-y)( x 3 -6)

Sample Set B

Identify the factors in each term.

Example 10

9 a 2 -6a-12 9 a 2 -6a-12 contains three terms. Some of the factors in each term are

first?term: 9?and? a 2 ,?or,?9?and?a?and?a second?term: -6?and?a third?term: -12?and?1,?or,?12?and?-1 first?term: 9?and? a 2 ,?or,?9?and?a?and?a second?term: -6?and?a third?term: -12?and?1,?or,?12?and?-1

Example 11

14 x 5 y+ (a+3) 2 14 x 5 y+ (a+3) 2 contains two terms. Some of the factors of these terms are

first?term: 14,? x 5 ,?y second?term: (a+3)?and?(a+3) first?term: 14,? x 5 ,?y second?term: (a+3)?and?(a+3)

Practice Set B

Exercise 5

In the expression 8 x 2 -5x+6 8 x 2 -5x+6 , list the factors of the
first term:
second term:
third term:

Solution

8, x x , x x ; -5 -5 , x x ; 6 and 1 or 3 and 2

Exercise 6

In the expression 10+2(b+6) (b-18) 2 10+2(b+6) (b-18) 2 , list the factors of the
first term:
second term:

Solution

10 and 1 or 5 and 2; 2, b+6 b+6 , b-18 b-18 , b-18 b-18

Common Factors

Common Factors

Sometimes, when we observe an expression carefully, we will notice that some particular factor appears in every term. When we observe this, we say we are observing common factors. We use the phrase common factors since the particular factor we observe is common to all the terms in the expression. The factor appears in each and every term in the expression.

Sample Set C

Name the common factors in each expression.

Example 12

5 x 3 -7 x 3 +14 x 3 5 x 3 -7 x 3 +14 x 3 .

The factor x 3 x 3 appears in each and every term. The expression x 3 x 3 is a common factor.

Example 13

4 x 2 +7x 4 x 2 +7x .

The factor x x appears in each term. The term 4 x 2 4 x 2 is actually 4xx 4xx . Thus, x x is a common factor.

Example 14

12x y 2 -9xy+15 12x y 2 -9xy+15 .

The only factor common to all three terms is the number 3. (Notice that 12=3·4,?9=3·3,?15=3·5 12=3·4,?9=3·3,?15=3·5 .)

Example 15

3(x+5)-8(x+5) 3(x+5)-8(x+5) .

The factor (x+5) (x+5) appears in each term. So, (x+5) (x+5) is a common factor.

Example 16

45 x 3 (x-7) 2 +15 x 2 (x-7)-20 x 2 (x-7) 5 45 x 3 (x-7) 2 +15 x 2 (x-7)-20 x 2 (x-7) 5 .

The number 5, the x 2 x 2 , and the (x-7) (x-7) appear in each term. Also, 5 x 2 (x-7) 5 x 2 (x-7) is a factor (since each of the individual quantities is joined by a multiplication sign). Thus, 5 x 2 (x-7) 5 x 2 (x-7) is a common factor.

Example 17

10 x 2 +9x-4 10 x 2 +9x-4 .

There is no factor that appears in each and every term. Hence, there are no common factors in this expression.

Practice Set C

List, if any appear, the common factors in the following expressions.

Exercise 7

x 2 +5 x 2 -9 x 2 x 2 +5 x 2 -9 x 2

Solution

x 2 x 2

Exercise 8

4 x 2 -8 x 3 +16 x 4 -24 x 5 4 x 2 -8 x 3 +16 x 4 -24 x 5

Solution

4 x 2 4 x 2

Exercise 9

4 (a+1) 3 +10(a+1) 4 (a+1) 3 +10(a+1)

Solution

2(a+1) 2(a+1)

Exercise 10

9ab(a-8)-15a (a-8) 2 9ab(a-8)-15a (a-8) 2

Solution

3a(a-8) 3a(a-8)

Exercise 11

14 a 2 b 2 c(c-7)(2c+5)+28c(2c+5) 14 a 2 b 2 c(c-7)(2c+5)+28c(2c+5)

Solution

14c(2c+5) 14c(2c+5)

Exercise 12

6( x 2 - y 2 )+19x( x 2 + y 2 ) 6( x 2 - y 2 )+19x( x 2 + y 2 )

Solution

no common factor

Coefficients

Coefficient

In algebra, as we now know, a letter is often used to represent some quantity. Suppose we represent some quantity by the letter x x . The notation 5x 5x means x+x+x+x+x x+x+x+x+x . We can now see that we have five of these quantities. In the expression 5x 5x , the number 5 is called the numerical coefficient of the quantity x x . Often, the numerical coefficient is just called the coefficient. The coefficient of a quantity records how many of that quantity there are.

Sample Set D

Example 18

12x 12x means there are 12x's 12x's .

Example 19

4ab 4ab means there are four ab's ab's .

Example 20

10(x-3) 10(x-3) means there are ten (x-3)'s (x-3)'s .

Example 21

1y 1y means there is one y y . We usually write just y y rather than 1y 1y since it is clear just by looking that there is only one y y .

Example 22

7 a 3 7 a 3 means there are seven a 3' s a 3' s .

Example 23

5ax 5ax means there are five ax's ax's . It could also mean there are 5ax's 5ax's . This example shows us that it is important for us to be very clear as to which quantity we are working with. When we see the expression 5ax 5ax we must ask ourselves "Are we working with the quantity ax ax or the quantity x x ?".

Example 24

6 x 2 y 9 6 x 2 y 9 means there are six x 2 y 9' s x 2 y 9' s . It could also mean there are 6 x 2 y 9' s 6 x 2 y 9' s . It could even mean there are 6 y 9 x 2' s 6 y 9 x 2' s .

Example 25

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

Practice Set D

Exercise 13

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 = 4x coefficient?is?4 x+x+x+x ? 4?terms = 4x coefficient?is?4
Exponents record the number of like factors in a term.
 x·x·x·x ? 4?factors = x 4 exponent?is?4 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.

Solution

how many of that quantity there are

Sample Set E

Example 26

3x 3x .

The coefficient of x x is 3.

Example 27

6 a 3 6 a 3 .

The coefficient of a 3 a 3 is 6.

Example 28

9(4-a) 9(4-a) .

The coefficient of (4-a) (4-a) is 9.

Example 29

3 8 x y 4 3 8 x y 4 .

The coefficient of x y 4 x y 4 is 3 8 3 8 .

Example 30

3 x 2 y 3 x 2 y .

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

Example 31

4 (x+y) 2 4 (x+y) 2 .

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

Practice Set E

Exercise 14

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

Solution

(a) 6 (b) x 3 x 3

Exercise 15

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

Solution

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

Exercise 16

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

Solution

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

Exercises

Exercise 17

What is an algebraic expression?

Solution

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

Exercise 18

Why is the number 14 considered to be an expression?

Exercise 19

Why is the number x x considered to be an expression?

Solution

x 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.

Exercise 20

2x+1 2x+1

Exercise 21

6x-10 6x-10

Solution

two:??6x,-10 two:??6x,-10

Exercise 22

2 x 3 +x-15 2 x 3 +x-15

Exercise 23

5 x 2 +6x-2 5 x 2 +6x-2

Solution

three:??5 x 2 ,6x,-2 three:??5 x 2 ,6x,-2

Exercise 24

3x 3x

Exercise 25

5cz 5cz

Solution

one:??5cz one:??5cz

Exercise 26

2

Exercise 27

61

Solution

one:??61 one:??61

Exercise 28

x x

Exercise 29

4 y 3 4 y 3

Solution

one:??4 y 3 one:??4 y 3

Exercise 30

17a b 2 17a b 2

Exercise 31

a+1 a+1

Solution

two:??a,1 two:??a,1

Exercise 32

(a+1) (a+1)

Exercise 33

2x+x+7 2x+x+7

Solution

three:??2x,x,7 three:??2x,x,7

Exercise 34

2x+(x+7) 2x+(x+7)

Exercise 35

(a+1)+(a-1) (a+1)+(a-1)

Solution

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

Exercise 36

a+1+(a-1) a+1+(a-1)

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

Exercise 37

x 2 +5 x 2 -2 x 2 x 2 +5 x 2 -2 x 2

Solution

x 2 x 2

Exercise 38

11 y 3 -33 y 3 11 y 3 -33 y 3

Exercise 39

45a b 2 +9 b 2 45a b 2 +9 b 2

Solution

9 b 2 9 b 2

Exercise 40

6 x 2 y 3 +18 x 2 6 x 2 y 3 +18 x 2

Exercise 41

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

Solution

( a+b ) ( a+b )

Exercise 42

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

Exercise 43

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

Solution

2a b 2 c 2 2a b 2 c 2

Exercise 44

4 x 2 y+5 a 2 b 4 x 2 y+5 a 2 b

Exercise 45

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

Solution

( a-3 ) ( a-3 )

Exercise 46

15 x 2 -30x y 2 15 x 2 -30x y 2

Exercise 47

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

Solution

no commom factors

Exercise 48

0.06a b 2 +0.03a 0.06a b 2 +0.03a

Exercise 49

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

Solution

( a+7 ) ( a+7 )

Exercise 50

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

Exercise 51

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

Solution

9 32 9 32

For the following problems, note how many:

Exercise 52

a's?in?4a? a's?in?4a?

Exercise 53

z's?in?12z? z's?in?12z?

Solution

12

Exercise 54

x 2 's?in?5 x 2 ? x 2 's?in?5 x 2 ?

Exercise 55

y 3 's?in?6 y 3 ? y 3 's?in?6 y 3 ?

Solution

6

Exercise 56

xy's?in?9xy? xy's?in?9xy?

Exercise 57

a 2 b's?in?10 a 2 b? a 2 b's?in?10 a 2 b?

Solution

10

Exercise 58

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

Exercise 59

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

Solution

8

Exercise 60

y 2 's?in?3 x 3 y 2 ? y 2 's?in?3 x 3 y 2 ?

Exercise 61

12x's?in?12 x 2 y 5 ? 12x's?in?12 x 2 y 5 ?

Solution

x y 5 x y 5

Exercise 62

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

Exercise 63

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

Solution

5x 5x

Exercise 64

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

Exercise 65

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

Solution

x 2 ( x-7 ) x 2 ( x-7 )

Exercise 66

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

Exercise 67

7y;?y 7y;?y

Solution

7

Exercise 68

10x;?x 10x;?x

Exercise 69

5a;?5 5a;?5

Solution

a a

Exercise 70

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

Exercise 71

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

Solution

6 x 2 b 2 6 x 2 b 2

Exercise 72

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

Exercise 73

9 a 2 b 5 ;?3a b 3 9 a 2 b 5 ;?3a b 3

Solution

3a b 2 3a b 2

Exercise 74

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

Exercise 75

( -4 ) a 6 b 2 ;?ab ( -4 ) a 6 b 2 ;?ab

Solution

( -4 ) a 5 b ( -4 ) a 5 b

Exercise 76

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

Exercises for Review

Exercise 77

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

Solution

16 x 16 ( x-1 ) 12 16 x 16 ( x-1 ) 12

Exercise 78

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

Exercise 79

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

Solution

-50 -50

Exercise 80

((Reference)) Find the value of 2 5 - 4 2 3 -2 2 5 - 4 2 3 -2 .

Exercise 81

((Reference)) Express 0.0000152 0.0000152 using scientific notation.

Solution

1.52× 10 -5 1.52× 10 -5

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