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4.2: Algebraic Expressions

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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 4.2.1

x+4 is an expression

Example 4.2.2

7y is an expression

Example 4.2.3

x3x2y7+9x is an expression.

Example 4.2.4

The number 8 is an expression. 8 can be written with explicit signs of operation by writing it as 8+0 or 81.

3x2+6=4x1 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 ab=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 4.2.5

3x4+6x2+5x+8

The expression has four terms: 3x4,6x2,5x,8.

Example 4.2.6

15y8

In this expression there is only one term. The term is 15y8.

Example 4.2.7

14x5y+(a+3)2.

In this expression there are two terms: the terms are 14x5y and (a+3)2. Notice that the term (a+3)2 is itself composed of two like factors, each of which is composed of the two terms, a and 3.

Example 4.2.8

m33

Using our definition of subtraction, this expression can be written in the form m3+(3). Now we can see that the terms are m3 and 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 4.2.9

p47p32p11.

Associating the sign with the individual quantities we see that the terms of this expression are p4, 7p3, 2p, and 11.

Practice Set A

Practice Problem 4.2.1

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

Answer

addition, multiplication

List the terms in the following expressions.
Practice Problem 4.2.2

4x28x+7

Answer

4x2,8x,7

Practice Problem 4.2.3

2xy+6x2+(xy)4

Answer

2xy,6x2,(xy)4

Practice Problem 4.2.4

5x2+3x3xy7+(xy)(x36)

Answer

5x2,3x,3xy7,(xy)(x36)

Sample Set B

Identify the factors in each term.

Example 4.2.10

9a26a12 contains three terms. Some of the factors in each term are

first term: 9 and a2, or 9 and a and a

second term: 6 and a

third term: 12 and 1, or, 12 and 1

Example 4.2.10

14x5y+(a+3)2 contains two terms. Some of the factors of these terms are

first term: 14,x5,y

second term: (a+3) and (a+3)

Practice Set B

Practice Problem 4.2.5

In the expression 8x25x+6, list the factors of the:

first term:
second term:
third term:

Answer

8,x,x

5,x

6 and 1 or 3 and 2

Practice Problem 4.2.6

In the expression 10+2(b+6)(b18)2, list the factors of the:

first term:
second term:
third term:

Answer

10 and 1 or 5 and 2

2,b+6,b18,b18

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 4.2.11

5x37x3+14x3

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

Example 4.2.12

4x2+7x

The factor x appears in each term. The term 4x2 is actually 4xx. Thus, x is a common factor.

Example 4.2.13

12xy29xy+15

The only factor common to all three terms is the number 3. (Notice that 12=34,9=33,15=35.

Example 4.2.14

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

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

Example 4.2.15

45x3(x7)2+15x2(x7)20x2(x7)5.

The number 5, the x2, and the (x7) appear in each term. Also, 5x2(x7) is also a factor (since each of the individual quantities is joined by a multiplication sign). Thus, 5x2(x7) is a common factor.

Example 4.2.16

10x2+9x4

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.

Practice Problem 4.2.7

x2+5x29x2

Answer

x2

Practice Problem 4.2.8

4x28x3+16x424x5

Answer

4x2

Practice Problem 4.2.9

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

Answer

2(a+1)

Practice Problem 4.2.10

9ab(a8)15a(a8)2

Answer

3a(a8)

Practice Problem 4.2.11

14a2b2c(c7)(2c+5)+28c(2c+5)

Answer

14c(2c+5)

Practice Problem 4.2.12

6(x2y2)+19x(x2+y2)

Answer

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. The notation 5x means x+x+x+x+x. We can now see that we have five of these quantities. In the expression 5x, the number 5 is called the numerical coefficient of the quantity 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 4.2.17

12x means that there are 12x's.

Example 4.2.18

4ab means there are four ab's.

Example 4.2.19

10(x3) means there are ten x3)'s.

Example 4.2.20

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

Example 4.2.21

7a3 means there are seven a3's.

Example 4.2.22

5ax means there are five ax's. It could also mean there are 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 we must ask ourselves "Are we working with the quantity ax or the quantity x?".

Example 4.2.23

6x2y9 means there are six x2y9's. It could also mean there are 6x2y9's. It could even mean there are 6y9x2's.

Example 4.2.24

5x3(y7) means there are five x3(y7)'s. It could also mean there are 5x3(x7)'s. It could also mean there are 5(x7)x3's.

Practice Set D

Practice Problem 4.2.13

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+x4 terms =4xcoefficient is 4
Exponents record the number of like factors in a term. xxxx4 factors =x4 exponent is 4
In a term, the coefficient of a particular group of factors is the remaining group of factors.

Answer

How many of that quantity there are.

Sample Set E

Example 4.2.25

3x

The coefficient of x is 3.

Example 4.2.26

6a3

The coefficient of a3 is 6.

Example 4.2.27

9(4a)

The coefficient of (4a) is 9.

Example 4.2.28

38xy4.

The coefficient of xy4 is 38

Example 4.2.25

3x2y.

The coefficient of x2y is 3; the coefficient of y is 3x2; and the coefficient of 3 is x2y.

Example 4.2.25

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

Practice Problem 4.2.14

Determine the coefficients.
In the term 6x3, the coefficient of:

(a) x3 is.

(b) 6 is.

Answer

(a) 6

(b) x3

Practice Problem 4.2.15

In the term 3x(y1), the coefficient of

(a) x(y1) is.

(b) (y1) is.

(c) 3(y1) is.

(d) x is.

(e) 3 is .

(f) The numerical coefficient is .

Answer

(a) 3

(b) 3x

(c) x

(d) 3(y1)

(e) x(y1)

(f) 3

Practice Problem 4.2.16

In the term 10ab4, the coefficient of

(a) ab4 is .

(b) b4 is .

(c) a is .

(d) 10 is .

(e) 10ab3 is .

Answer

(a) 10

(b) 10a

(c) 10b4

(d) ab4

(e) b

Exercises

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

Exercise 4.2.1

What is an algebraic expression?

Answer

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

Exercise 4.2.2

Why is the number 14 considered to be an expression?

Exercise 4.2.3

Why is the number x considered to be an expression?

Answer

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

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

Exercise 4.2.4

2x+1

Exercise 4.2.5

6x10

Answer

two

6x, 10

Exercise 4.2.6

2x3+x15

Exercise 4.2.7

5x2+6x2

Answer

three

5x2,6x,2

Exercise 4.2.8

3x

Exercise 4.2.9

5cz

Answer

one

5cz

Exercise 4.2.10

2

Exercise 4.2.11

61

Answer

one

61

Exercise 4.2.12

x

Exercise 4.2.13

4y3

Answer

one

4y3

Exercise 4.2.14

17ab2

Exercise 4.2.15

a+1

Answer

two

a,1

Exercise 4.2.16

(a+1)

Exercise 4.2.17

2x+x+7

Answer

three

2x,x,7

Exercise 4.2.18

2x+(x+7)

Exercise 4.2.19

(a+1)+(a1)

Answer

two

(a+b),(a1)

Exercise 4.2.20

a+1+(a1)

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

Exercise 4.2.21

x2+5x22x2

Answer

x2

Exercise 4.2.22

11y333y3

Exercise 4.2.23

45ab2+9b2

Answer

9b2

Exercise 4.2.24

6x2y3+18x2

Exercise 4.2.25

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

Answer

(a+b)

Exercise 4.2.26

14ab2c2(c+8)+12ab2c2

Answer

2ab2c2

Exercise 4.2.27

4x2y+5a2b

Exercise 4.2.28

9a(a3)2+10b(a3)

Answer

(a3)

Exercise 4.2.29

15x230xy2

Exercise 4.2.30

12a3b2c7(b+1)(ca)

Answer

no common factors

Exercise 4.2.31

0.06ab2+0.03a

Exercise 4.2.32

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

Answer

(a+7)

Exercise 4.2.33

34x2y2z2+38x2z2

Exercise 4.2.34

916(a2b2)+932(b2a2)

Answer

932

For the following problems, note how many:

Exercise 4.2.35

a's in 4a?

Exercise 4.2.36

z's in 12z?

Answer

12

Exercise 4.2.37

x2's in 5x2?

Exercise 4.2.38

y3's in 6y3?

Answer

6

Exercise 4.2.39

xy's in 9xy?

Exercise 4.2.40

a2b's in 10a2b?

Answer

10

Exercise 4.2.41

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

Exercise 4.2.42

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

Answer

8

Exercise 4.2.43

y2's in 3x3y2?

Exercise 4.2.44

12x's in 12x2y5?

Answer

xy5

Exercise 4.2.45

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

Exercise 4.2.46

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

Answer

5x

Exercise 4.2.47

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

Exercise 4.2.48

2's in 2x2(x7)?

Answer

x2(x7)

Exercise 4.2.49

3(a+8)'s in 6x2(a+8)3(a8)?

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 4.2.50

7y;y

Answer

7

Exercise 4.2.51

10x;x

Exercise 4.2.52

5a;5

Answer

a

Exercise 4.2.53

12a2b3c2r7;a2c2r7

Exercise 4.2.54

6x2b2(c1);c1

Answer

6x2b2

Exercise 4.2.55

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

Exercise 4.2.56

9a2b5;3ab3

Answer

3ab

Exercise 4.2.57

15x4y4(z+9a)3;(z+9a)

Exercise 4.2.58

(4)a6b2;ab

Answer

(4)a5b

Exercise 4.2.59

(11a)(a+8)3(a1);(a+8)2

Exercises for Review

Exercise 4.2.60

Simplify [2x8(x1)5x4(x1)2]4

Answer

16x16(x1)12

Exercise 4.2.61

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

Exercise 4.2.62

Find the value of [6(42)+7(3+5)].

Answer

50

Exercise 4.2.63

Find the value of 254232

Exercise 4.2.64

Express 0.0000152 using scientific notation.

Answer

1.52×105


This page titled 4.2: Algebraic Expressions is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Denny Burzynski & Wade Ellis, Jr. (OpenStax CNX) .

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