Fundamentals of Algebra Copyright © Cengage Learning. All rights reserved.

2.2 Simplifying Algebraic Expressions

What You Will Learn Use the properties of algebra. Combine like terms of an algebraic expression. Simplify an algebraic expression by rewriting the terms. Use the Distributive Property to remove symbols of grouping.

Properties of Algebra

Review of Properties of Real Numbers Properties of Algebra You are now ready to combine algebraic expressions using the properties of real numbers. Review of Properties of Real Numbers 1. Commutative Property of Addition: Two real numbers can be added in either order. a + b = b + a 3 + 5 = 5 + 3 2. Commutative Property of Multiplication: Two real numbers can be multiplied in either order. ab = ba 4 •(-7) = -7 • 4

Properties of Algebra 3. Associative Property of Addition: When three real numbers are added, it makes no different which two are added first. (a + b) + c = a + (b + c) (2 + 6) + 5 = 2 + (6 + 5) 4. Associative Property of Multiplication: When three real numbers are multiplied, it makes no difference which two are multiplied first. (ab)c = a(bc) (3 • 5) • 2 = 3 • (5 • 2) 5. Distributive Property: Multiplication distributes over addition. a(b + c) = ab + ac 3(8 + 5) = 3 • 8 + 3 • 5 (a + b)c = ac + bc (3 + 8)5 = 3 • 5 + 8 • 5

Properties of Algebra 6. Additive Identity Property: The sum of zero and a real number equals the number itself. a + 0 = 0 + a = a 3 + 0 = 0 + 3 = 3 7. Multiplicative Identity Property: The product of 1 and a real number equals the number itself a • 1 = 1 • a = a 4 • 1 = 1 • 4 = 4 8. Additive Inverse Property: The sum of a real number and its opposite is zero a + (-a) = 0 3 + (-3) = 0 9. Multiplicative Inverse Property: The product of a nonzero real number and it’s reciprocal is 1

Example 1 – Applying Properties of Real Numbers Use the indicated rule to complete each statement. a. Additive Identity Property: (x – 2) + = x – 2 b. Commutative Property of Multiplication: 5(y + 6) = c. Commutative Property of Addition: 5(y + 6) = d. Distributive Property: 5(y + 6) = e. Associative Property of Addition: (x2 + 3) + 7 = f. Additive Inverse Property: + 4x = 0

Example 1 – Applying Properties of Real Numbers cont’d a. (x – 2) + 0 = x – 2 b. 5(y + 6) = (y + 6)5 c. 5(y + 6) = 5(6 + y) d. 5(y + 6) = 5y + 5(6) e. (x2 + 3) + 7 = x2 + (3 + 7) f. –4x + 4x = 0

Example 2 – Identifying the Terms of Expressions Use the Distributive Property to expand each expression. a. 2(7 – x) b. (10 – 2y)3 c. 2x(x + 4y) d. –(1 – 2y + x) Solution: a. 2(7 – x) = 2  7 – 2  x = 14 – 2x b. (10 – 2y)3 = 10(3) – 2y(3) = 30 – 6y

Example 2 – Identifying the Coefficients of Terms cont’d c. 2x(x + 4y) = 2x(x) + 2x(4y) = 2x2 + 8xy d. –(1 – 2y + x) = (–1)(1 – 2y + x) = (–1)(1) – (–1)(2y) + (–1)(x) = –1 + 2y – x

Example 3 – Geometry: Visualizing the Distributive Property Write the area of each component of each figure. Then demonstrate the Distributive Property by writing the total area of each figure in two ways.

Example 3 – Geometry: Visualizing the Distributive Property cont’d The total area is 3(2 + 4) = 3 • 2 + 3 • 4 = 6 + 12 = 18 6 12 The total area is a(a + b) = a • a + a • b = a2 + ab a2 ab

Example 4 – Using Mental Math in Everyday Applications You can earn $14 per hour and time-and-a-half for overtime. Show how you can use the Distributive property to find your overtime wage mentally. Solution: You can think of the “time-and-a-half” as . So, using the Distributive Property, you can think of the following without writing anything on paper.

Example 4 – Using Mental Math in Everyday Applications cont’d b. You are buying 15 potted plants that cost $19 each. Show how you can use the Distributive Property to find the total cost mentally. Solution: b. You can think of the $19 as $20 – $1. So, using the Distributive Property, you can think of the following without writing anything on paper. 15(19) = 15(20 – 1) = 15(20) – 15(1) = 300 – 15 = $285

Combining Like Terms

Combining Like Terms In an algebraic expression, two terms are said to be like terms if they are both constant terms or if they have the same variable factor(s). Factors such as x and 5x and ab in 6ab are called variable factors. The terms 5x and –3x are like terms because they have the same variable factor, x. Similarly, 3x2y, –x2y, and (x2y) are like terms because they have the same variable factors, x2 and y.

Example 5 – Identifying Like Terms Expression Like Terms a. 5xy + 1 – xy 5xy and –xy b. 12 – x2 + 3x – 5 12 and –5 c. 7x – 3 – 2x + 5 7x and –2x, –3 and 5

Combining Like Terms To combine like terms in an algebraic expression, you can simply add their respective coefficients and attach the common variable factor(s). This is actually an application of the Distributive Property, as shown in Example 5.

Example 6 – Combining Like Terms Simplify each expression by combining like terms. a. 5x + 2x – 4 b. –5 + 8 + 7y – 5y c. 2y – 3x – 4x d. 7x + 3y – 4x e. 12a – 5 – 3a + 7 Solution a. 5x + 2x – 4 = (5 + 2)x – 4 = 7x – 4 b. –5 + 8 + 7y – 5y = (–5 + 8) + (7 – 5)y = 3 + 2y Distributive Property Add. Distributive Property Simplify.

Example 6 – Combining Like Terms cont’d c. 2y – 3x – 4x = 2y – x(3 + 4) = 2y – x(7) = 2y – 7x d. 7x + 3y – 4x = 3y + 7x – 4x = 3y + (7x – 4x) = 3y + (7 – 4)x = 3y + 3x e. 12a – 5 – 3a + 7 = 12a – 5 – 3a + 7 = (12a – 3a) + (–5 +7) = (12 – 3)a + (–5 +7) = 9a + 12 Distributive Property Add. Simplify. Commutative Property Associative Property Distributive Property Subtract Commutative Property Associative Property Distributive Property Simplify

Simplifying Algebraic Expressions

Simplifying Algebraic Expressions Simplifying an algebraic expression by rewriting it in a more usable form is one of the most frequently used skills in algebra. To simplify an algebraic expression generally means to remove symbols of grouping and combine like terms. For instance, the expression x + (3 + x) can be simplified as 2x + 3.

Example 7 – Simplifying Algebraic Expressions Simplify each expression. a. –3(–5x) b. 7(–x) c. d. x2(–2x3) e. (–2x)(4x) f. (2rs)(r2s) Solution a. –3(–5x) = (–3)(–5)x = 15x b. 7(–x) = 7(–1)(x) = –7x Associative Property Multiply. Coefficient of –x is –1. Multiply.

Example 7 – Simplifying Algebraic Expressions cont’d Solution c. = 1 • x = x d. x2(–2x3) = (–2)(x2 • x3) = –2 • x • x • x •x • x = –2x5 Coefficient of Commutative and Associative Properties Multiplicative Inverse Multiplicative Identity Commutative and Associative Properties Repeated Multiplication Exponential Form

Example 7 – Simplifying Algebraic Expressions cont’d Commutative and Associative Properties e. (–2x)(4x) = (–2 • 4)(x •x) = –8x2 f. (2rs)(r2s) = 2(r • r2)(s • s) = 2 • r • r • r • s • s = 2r3s2 Exponential Form Commutative and Associative Properties Repeated multiplication Exponential Form

Symbols of Grouping

Example 8 – Removing Symbols of Grouping The main tool for removing symbols of grouping is the Distributive Property, as illustrated in this example. Simplify each expression. a. –(3y + 5) b. 5x + (x – 7)2 c. –2(4x – 1) + 3x d. 3(y – 5) – (2y – 7)

Example 8 – Removing Symbols of Grouping cont’d a. –(3y + 5) = –3y – 5 b. 5x + (x – 7)2 = 5x + 2x – 14 = 7x – 14 c. –2(4x – 1) + 3x = –8x + 2 + 3x = –8x + 3x + 2 = –5x + 2 d. 3(y – 5) – (2y – 7) = 3y – 15 – 2y + 7 = (3y – 2y) + (–15 + 7) = y – 8 Distributive Property Distributive Property Combine like terms. Distributive Property Commutative Property Combine like terms. Distributive Property Group like terms. Combine like terms.

Example 9 – Geometry: Writing and Simplifying a Formula Write and simplify an expression for (a) the perimeter and (b) the area of the triangle.

Example 9 – Geometry: Writing and Simplifying a Formula cont’d Solution: a. Perimeter of a Triangle = Sum of the Three Sides = 2x + (2x + 4) + (x + 5) = (2x + 2x + x) + (4 + 5) = 5x + 9 b. Area of a Triangle = • Base • Height Substitute Group like terms Combine like terms Substitute Commutative Property Multiply Distributive Property

Example 10 – Geometry: Writing and Simplifying a Formula The formula for the area of a trapezoid is . Use this formula to write and simplify an expression for the area of the proposed trapezoidal park.

Example 10 – Geometry: Writing and Simplifying a Formula cont’d Solution: Begin by assigning the follow values. h = x mi b1 = (x + 0.5) mi b2 = (3x + .03) mi Then use the formula to write an expression.