11.1: Algebraic Expressions

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Learning Objectives

be able to recognize an algebraic expression
be able to distinguish between terms and factors
understand the meaning and function of coefficients
be able to perform numerical evaluation

Algebraic Expressions

Numerical Expression
In arithmetic, a numerical expression results when numbers are connected by arithmetic operation signs (+, -, ⋅ , ÷). For example, \(8 + 5\), \(4 - 9\), \(3 \cdot 8\), and \(9 \div 7\) are numerical expressions.

Algebraic Expression
In algebra, letters are used to represent numbers, and an algebraic expression results when an arithmetic operation sign associates a letter with a number or a letter with a letter. For example, \(x + 8\), \(4 - y\), \(3 \cdot x\), \(x \div 7\), and \(x \cdot y\) are algebraic expressions.

Expressions
Numerical expressions and algebraic expressions are often referred to simply as expressions.

Terms and Factors

In algebra, it is extremely important to be able to distinguish between terms and factors.

Distinction Between Terms and Factors
Terms are parts of sums and are therefore connected by + signs.
Factors are parts of products and are therefore separated by \(\cdot\) signs.

Note

While making the distinction between sums and products, we must remember that subtraction and division are functions of these operations.

In some expressions it will appear that terms are separated by minus signs. We must keep in mind that subtraction is addition of the opposite, that is,
\(x - y = x + (-y)\)
In some expressions it will appear that factors are separated by division signs. We must keep in mind that
\(\dfrac{x}{y} = \dfrac{x}{1} \cdot \dfrac{1}{y} = x \cdot \dfrac{1}{y}\)

Sample Set A

State the number of terms in each expression and name them.

\(x + 4\). In this expression, \(x\) and 4 are connected by a "+" sign. Therefore, they are terms. This expression consists of two terms.

Sample Set A

\(y - 8\). The expression \(y - 8\) can be expressed as \(y + (-8)\). We can now see that this expression consists of the two terms \(y\) and -8.

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

Sample Set A

\(a + 7 - b - m\). Associating the sign with the individual quantities, we see that this expression consists of the four terms \(a\), 7, \(-b\), \(-m\).

Sample Set A

\(5m - 8n\). This expression consists of the two terms, \(5m\) and \(-8n\). Notice that the term \(5m\) is composed of the two factors 5 and \(m\). The term \(-8n\) is composed of the two factors -8 and \(n\).

Sample Set A

\(3x\). This expression consists of one term. Notice that \(3x\) can be expressed as \(3x + 0\) or \(3x \cdot 1\) (indicating the connecting signs of arithmetic). Note that no operation sign is necessary for multiplication.

Practice Set A

Specify the terms in each expression.

\(x + 7\)

Answer: \(x, 7\)

Practice Set A

\(3m - 6n\)

Answer: \(3m - 6n\)

Practice Set A

\(5y\)

Answer: \(5y\)

Practice Set A

\(a + 2b - c\)

Answer: \(a, 2b, -c\)

Practice Set A

\(-3x - 5\)

Answer: \(-3x, -5\)

Coefficients

We know that multiplication is a description of repeated addition. For example,
\(5 \cdot 7\) describes \(7 + 7 + 7 + 7 + 7\)

Suppose some quantity is represented by the letter \(x\). The multiplication \(5x\) describes \(x + x + x + x + x\). It is now easy to see that \(5x\) specifies 5 of the quantities represented by \(x\). In the expression \(5x\), 5 is called the numerical coefficient, or more simply, the coefficient of \(x\).

Coefficient
The coefficient of a quantity records how many of that quantity there are.

Since constants alone do not record the number of some quantity, they are not usually considered as numerical coefficients. For example, in the expression \(7x + 2y - 8z + 12\), the coefficient of

\(7x\) is 7. (There are 7 \(x\)'s.)
\(2y\) is 2. (There are 2 \(y\)'s.)
\(-8z\) is -8. (There are -8 \(z\)'s.)

The constant 12 is not considered a numerical coefficient.

\(1x = x\).

When the numerical coefficient of a variable is 1, we write only the variable and not the coefficient. For example, we write \(x\) rather than \(1x\). It is clear just by looking at \(x\) that there is only one.

Numerical Evaluation

We know that a variable represents an unknown quantity. Therefore, any expression that contains a variable represents an unknown quantity. For example, if the value of \(x\) is unknown, then the value of \(3x + 5\) is unknown. The value of \(3x + 5\) depends on the value of \(x\).

Numerical Evaluation
Numerical evaluation is the process of determining the numerical value of an algebraic expression by replacing the variables in the expression with specified numbers.

Sample Set B

Find the value of each expression.

\(2x + 7y\), if \(x = -4\) and \(y = 2\).

Solution

Replace \(x\) with -4 and \(y\) with 2.

\(\begin{array} {rcl} {2x + 7y} & = & {2(-4) + 7(2)} \\ {} & = & {-8 + 14} \\ {} & = & {6} \end{array}\)

Thus, when \(x = -4\) and \(y = 2\), \(2x + 7y = 6\).

Sample Set B

\(\dfrac{5a}{b} + \dfrac{8b}{12}\), if \(a = 6\) and \(b = -3\).

Solution

Replace \(a\) with 6 and \(b\) with -3.

\(\begin{array} {rcl} {\dfrac{5a}{b} + \dfrac{8b}{12}} & = & {\dfrac{5(6)}{-3} + \dfrac{8(-3)}{12}} \\ {} & = & {\dfrac{30}{-3} + \dfrac{-24}{12}} \\ {} & = & {-10 + (-2)} \\ {} & = & {-12} \end{array}\)

Thus, when \(a = 6\) and \(b = -3\), \(\dfrac{5a}{b} + \dfrac{8b}{12} = -12\)

Sample Set B

\(6(2a - 15b)\), if \(a = -5\) and \(b = -1\).

Solution

Replace \(a\) with -5 and \(b\) with -1.

\(\begin{array} {rcl} {6(2a - 15b)} & = & {6(2(-5) - 15(-1))} \\ {} & = & {6(-10 + 15)} \\ {} & = & {6(5)} \\ {} & = & {30} \end{array}\)

Thus, when \(a = -5\) and \(b = -1\), \(6(2a - 15b) = 30\)

Sample Set B

\(3x^2 - 2x + 1\), if \(x = 4\)

Solution

Replace \(x\) with 4.

\(\begin{array} {rcl} {3x^2 - 2x + 1} & = & {3(4)^2 - 2(4) + 1} \\ {} & = & {3 \cdot 16 - 2(4) + 1} \\ {} & = & {48 - 8 + 1} \\ {} & = & {41} \end{array}\)

Thus, when \(x = 4\). \(3x^2 - 2x + 1 = 41.\)

Sample Set B

\(-x^2 - 4\), if \(x = 3\)

Solution

Replace \(x\) with 3.

\(\begin{array} {rcl} {-x^2 - 4} & = & {-3^2 - 4 \ \ \ \text{Be careful to square only the 3. The exponent 2 is connected only to 3, not -3}} \\ {} & = & {-9 - 4} \\ {} & = & {-13} \end{array}\)

Sample Set B

\((-x)^2 - 4\), if \(x = 3\)

Solution

Replace \(x\) with 3.

\(\begin{array} {rcl} {(-x)^2 - 4} & = & {(-3)^2 - 4 \ \ \ \text{The exponent is connected to -3, not 3 as in problem 5 above.}} \\ {} & = & {9 - 4} \\ {} & = & {5} \end{array}\)

The exponent is connected to –3, not 3 as in the problem above.

Pracitce Set B

Find the value of each expression.

\(9m - 2n\), if \(m = -2\) and \(n = 5\)

Answer: -28

Pracitce Set B

\(-3x - 5y + 2z\), if \(x = -4, y = 3, z = 0\)

Answer: -3

Pracitce Set B

\(\dfrac{10a}{3b} + \dfrac{4b}{2}\), if \(a = -6\), and \(b = 2\)

Answer: -6

Pracitce Set B

\(8(3m - 5n)\), if \(m = -4\) and \(n = -5\)

Answer: 104

Pracitce Set B

\(3[-40 - 2(4a - 3b)]\), if \(a = -6\) and \(b = 0\)

Answer: 24

Pracitce Set B

\(5y^2 + 6y - 11\), if \(y = -1\)

Answer: -12

Pracitce Set B

\(-x^2 + 2x + 7\), if \(x = 4\)

Answer: -1

Pracitce Set B

\((-x)^2 + 2x + 7\), if \(x = 4\)

Answer: 31

Exercises

Exercise \(\PageIndex{1}\)

In an algebraic expression, terms are separated by signs and factors are separated by signs.

Answer: Addition; multiplication

For the following 8 problems, specify each term.

Exercise \(\PageIndex{2}\)

\(3m + 7n\)

Exercise \(\PageIndex{3}\)

\(5x + 18y\)

Answer: \(5x, 18y\)

Exercise \(\PageIndex{4}\)

\(4a - 6b + c\)

Exercise \(\PageIndex{5}\)

\(8s + 2r - 7t\)

Answer: \(8s, 2r, -7t\)

Exercise \(\PageIndex{6}\)

\(m - 3n - 4a + 7b\)

Exercise \(\PageIndex{7}\)

\(7a - 2b - 3c - 4d\)

Answer: \(7a, -2b, -3c, -4d\)

Exercise \(\PageIndex{8}\)

\(-6a - 5b\)

Exercise \(\PageIndex{9}\)

\(-x - y\)

Answer: \(-x, -y\)

Exercise \(\PageIndex{10}\)

What is the function of a numerical coefficient?

Exercise \(\PageIndex{11}\)

Write \(1m\) in a simpler way.

Answer: \(m\)

Exercise \(\PageIndex{12}\)

Write 1s in a simpler way.

Exercise \(\PageIndex{13}\)

In the expression 5a, how many a’s are indicated?

Answer: 5

Exercise \(\PageIndex{14}\)

In the expression –7c, how many c’s are indicated?

Find the value of each expression.

Exercise \(\PageIndex{15}\)

\(2m - 6m\), if \(m = -3\) and \(n = 4\)

Answer: -30

Exercise \(\PageIndex{16}\)

\(5a + 6b\), if \(a = -6\) and \(b = 5\)

Exercise \(\PageIndex{17}\)

\(2x - 3y + 4z\), if \(x = 1\), \(y = -1\), and \(z = -2\)

Answer: -3

Exercise \(\PageIndex{18}\)

\(9a + 6b - 8x + 4y\), if \(a = -2\), \(b = -1\), \(x = -2\), and \(y = 0\)

Exercise \(\PageIndex{19}\)

\(\dfrac{8x}{3y} + \dfrac{18y}{2x}\), if \(x = 9\) and \(y = -2\)

Answer: -14

Exercise \(\PageIndex{20}\)

\(\dfrac{-3m}{2n} - \dfrac{-6n}{m}\), if \(m = -6\) and \(n = 3\)

Exercise \(\PageIndex{21}\)

\(4(3r + 2s)\), if \(r = 4\) and \(s = 1\)

Answer: 56

Exercise \(\PageIndex{22}\)

\(3(9a - 6b)\), if \(a = -1\) and \(b = -2\)

Exercise \(\PageIndex{23}\)

\(-8 (5m + 8n)\), if \(m = 0\) and \(n = -1\)

Answer: 64

Exercise \(\PageIndex{24}\)

\(-2(-6x + y - 2z)\), if \(x = 1, y = 1\), and \(z = 2\)

Exercise \(\PageIndex{25}\)

\(-(10x - 2y + 5z)\) if \(x = 2, y = 8\), and \(z = -1\)

Answer: 1

Exercise \(\PageIndex{26}\)

\(-(a - 3b + 2c - d)\), if \(a = -5, b = 2, c = 0\), and \(d = -1\)

Exercise \(\PageIndex{27}\)

\(3[16 - 3(a + 3b)]\), if \(a = 3\) and \(b = -2\)

Answer: 75

Exercise \(\PageIndex{28}\)

\(-2[5a + 2b(b - 6)]\), if \(a = -2\) and \(b = 3\)

Exercise \(\PageIndex{29}\)

\(-\{6x + 3y[-2(x + 4y)]\}\), if \(x = 0\) and \(y = 1\)

Answer: 24

Exercise \(\PageIndex{30}\)

\(-2\{19 - 6[4 - 2(a - b - 7)]\}\), if \(a = 10\) and \(b = 3\)

Exercise \(\PageIndex{31}\)

\(x^2 + 3x - 1\), if \(x = 5\)

Answer: 39

Exercise \(\PageIndex{32}\)

\(m^2 - 2m + 6\), if \(m = 3\)

Exercise \(\PageIndex{33}\)

\(6a^2 + 2a - 15\), if \(a = -2\)

Answer: 5

Exercise \(\PageIndex{34}\)

\(5s^2 + 6s + 10\), if \(x = -1\)

Exercise \(\PageIndex{35}\)

\(16x^2 + 8x - 7\), if \(x = 0\)

Answer: -7

Exercise \(\PageIndex{36}\)

\(-8y^2 + 6y + 11\), if \(y = 0\)

Exercise \(\PageIndex{37}\)

\((y - 6)^2 + 3(y - 5) + 4\), if \(y = 5\)

Answer: 5

Exercise \(\PageIndex{38}\)

\((x + 8)^2 + 4(x + 9) + 1\), if \(x = -6\)

Exercises for Review

Exercise \(\PageIndex{37}\)

Perform the addition: \(5 \dfrac{3}{8} + 2\dfrac{1}{6}\).

Answer: \(\dfrac{181}{24} = 7 \dfrac{13}{24}\)

Exercise \(\PageIndex{38}\)

Arrange the numbers in order from smallest to largest: \(\dfrac{11}{32}\), \(\dfrac{15}{48}\), and \(\dfrac{7}{16}\)

Exercise \(\PageIndex{37}\)

Find the value of \((\dfrac{2}{3})^2 + \dfrac{8}{27}\)

Answer: \(\dfrac{20}{27}\)

Exercise \(\PageIndex{38}\)

Write the proportion in fractional form: “9 is to 8 as x is to 7.”

Exercise \(\PageIndex{37}\)

Find the value of \(-3(2 - 6) - 12\)

Answer: 0