11.1: Algebraic Expressions
- 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.
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
\(\dfrac{x}{y} = \dfrac{x}{1} \cdot \dfrac{1}{y} = x \cdot \dfrac{1}{y}\)
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.
\(y - 8\). 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.
\(a + 7 - b - m\). Associating the sign with the individual quantities, we see that this expression consists of the four terms \(a\), 7, \(-b\), \(-m\).
\(5m - 8n\). 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 \(n\).
\(3x\). 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.
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 \(y = 2\), \(2x + 7y = 6\).
\(\dfrac{5a}{b} + \dfrac{8b}{12}\), 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 \(b = -3\), \(\dfrac{5a}{b} + \dfrac{8b}{12} = -12\)
\(6(2a - 15b)\), 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 \(b = -1\), \(6(2a - 15b) = 30\)
\(3x^2 - 2x + 1\), 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 \(x = 4\). \(3x^2 - 2x + 1 = 41.\)
\(-x^2 - 4\), if \(x = 3\)
Solution
Replace \(x\) with 3.
\(\begin{array} {rcl} {-x^2 - 4} & = & {-3^2 - 4 \ \ \ \text{Be careful to square only the 3. The exponent 2 is connected only to 3, not -3}} \\ {} & = & {-9 - 4} \\ {} & = & {-13} \end{array}\)
\((-x)^2 - 4\), 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 1 s in a simpler way.
Exercise \(\PageIndex{13}\)
In the expression 5 a , how many a ’s are indicated?
- Answer
-
5
Exercise \(\PageIndex{14}\)
In the expression –7 c , 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