Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.8 – Slide 1

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.8 – Slide 2 The Real Number System Chapter 1

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.8 – Slide Simplifying Expressions

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.8 – Slide 4 Objectives 1.Simplify expressions. 2.Identify terms and numerical coefficients. 3.Identify like terms. 4.Combine like terms. 5.Simplify expressions from word phrases. 1.8 Simplifying Expressions

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.8 – Slide 5 Example 1 Simplify each expression. (a) 2m – = 2m + 5 (b) –5(3x – 2y) =–5(3x) – (–5)(2y) 1.8 Simplifying Expressions Simplifying Expressions = –15x + 10y (c) –3 – (4p + 7) =–3 – 1(4p + 7) = –3 – 4p – 7 = –4p – 10 Note: We mentally used the commutative and associative properties to add in the last step. In practice, these steps are usually left out, but we should realize that they are used whenever the ordering and grouping in a sum are rearranged.

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.8 – Slide 6 A term is a number, variable, or a product or quotient of a number and one or more variables raised to powers. Examples of terms include: –9x 2, 15y, –3, 8m 2 n, and k. Identifying Terms and Numerical Coefficients 1.8 Simplifying Expressions The numerical coefficient, or simply coefficient, of the term 9m is 9; the numerical coefficient of –5x 3 y 2 is –15; the numerical coefficient of x is 1; and the numerical coefficient of 8 is 8. In the expression the numerical coefficient of x is since

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.8 – Slide 7 CAUTION It is important to be able to distinguish between terms and factors. For example, in the expression, there are two terms, and. Terms are separated by a + or – sign. On the other hand, in the one-term expression, and are factors. Factors are multiplied. Identifying Terms and Numerical Coefficients 1.8 Simplifying Expressions

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.8 – Slide 8 Terms with exactly the same variables (including the same exponents) are called like terms. For example, 9m and 4m have the same variables and are like terms. Also, 6x 3 and –5x 3 are like terms. The terms –4y 3 and 4y 2 have different exponents and are unlike terms. Here are some additional examples: 5x and –12x 3x 2 y and 5x 2 y Like terms 4xy 2 and 5xy 8x 2 y 3 and 7x 3 y 2 Unlike terms Identifying Like Terms 1.8 Simplifying Expressions

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.8 – Slide 9 Recall the distributive property: x(y + z) = xy + xz. As seen in the previous section, this statement can also be written as xy + xz = x(y + z) or yx + zx = (y + z)x. Thus, the distributive property may be used to find the sum or difference of like terms. For example, 3x + 5x = (3 + 5)x = 8x. This process is called combining like terms. Combining Like Terms 1.8 Simplifying Expressions

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.8 – Slide 10 Example 2 Combine like terms in each expression. (a) 12x + 8x = (12 + 8)x = (b) –2y + 5y – 9y =(–2 + 5 – 9)y = (b) –3x 2 y + 4xy 2 =Cannot be combined because these are unlike terms. Combining Like Terms 1.8 Simplifying Expressions 20x –6y

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.8 – Slide 11 Example 3 Simplify each expression. (a) 2a – 5 – 3(a + 1) =2a – 5 – 3a – 3 Combining Like Terms 1.8 Simplifying Expressions = –a – 8 (b) 4b(2b + 5) – 2(3b – 1) =8b b – 6b + 2 = 8b b + 2

Copyright © 2010 Pearson Education, Inc. All rights reserved. 1.8 – Slide 12 Example 4 Write the following phrase as a mathematical expression and simplify. Four times a number, subtracted from the sum of twice the number and 4. (2x + 4)2x + 4 – 4x = Simplifying Expressions from Word Phrases 1.8 Simplifying Expressions Four times the number The sum of twice the number and 4 – 4x = –2x + 4