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**Columbus State Community College**

Chapter 2 Section 2 Simplifying Expressions

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**Simplifying Expressions**

Combine like terms, using the distributive property. Simplify expressions. Use the distributive property to multiply.

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**Simplifying Expressions**

The basic idea in simplifying expressions is to combine like terms through addition and subtraction. Each addend and subtrahend in an expression is a term. Separates the terms x y – 5 1 Three terms x 4y 5 is called a variable term. The coefficient is 1 and the variable is x. is called a variable term. The coefficient is 4 and the variable is y. ( or –5 ) is called a constant term.

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Like Terms Like Terms Like terms are terms with exactly the same variable parts (the same letters and exponents). The coefficients do not have to match.

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**–6a3c5 and 2a3c5 Examples of Like Terms Like Terms 1. 2x and –4x**

Variable parts match; both are x. 2. 8b7 and 2b7 Variable parts match; both are b7. 3. –6a3c5 and 2a3c5 Variable parts match; both are a3c5. 4. 4 and –5 No variable parts; numbers are like terms.

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**Examples of Unlike Terms**

Variable parts do not match; exponents are different. 1. 4x and –2x2 Variable parts do not match; letters are different. 2. 3a and 3b Variable parts do not match; exponents are different. 3. m2n3 and m2n5 Variable parts do not match; one term has a variable part, but the other term does not. 4. 7v and –5

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**Identifying Like Terms and Their Coefficients**

EXAMPLE Identifying Like Terms and Their Coefficients List the like terms in each expression. Then identify the coefficients of the like terms. (a) x + –7x2 + –4xy + –5x – 8 The like terms are x and –5x. The coefficient of x is understood to be 1, and the coefficient of –5x is –5.

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**Identifying Like Terms and Their Coefficients**

EXAMPLE Identifying Like Terms and Their Coefficients List the like terms in each expression. Then identify the coefficients of the like terms. (b) 3a2b + 2ab2 – 4a2b + 8a2b2 – a2b The like terms are 3a2b, 4a2b, and a2b. The coefficient of 3a2b is 3, the coefficient of 4a2b is 4 (or –4), and the coefficient of a2b is 1 (or –1).

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**Identifying Like Terms and Their Coefficients**

EXAMPLE Identifying Like Terms and Their Coefficients List the like terms in each expression. Then identify the coefficients of the like terms. (c) 8m n + 6mn + 7mn2 – 5 The like terms are 2 and 5 (or –5). The like terms are constants (there are no variable parts).

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**Variable part is unchanged.**

CAUTION CAUTION Notice that 4x and 7x is simplified to 11x, not to 11x2. Variable part is unchanged. Do not change x to x2.

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Combining Like Terms Combining Like Terms Step 1 If there are any variable terms with no coefficient, write in the understood 1. Step 2 If there are any subtractions, change each one to adding the opposite. Alternatively, you can treat each term that follows a subtraction operator as a negative term. Step 3 Find like terms (the variable parts match). Step 4 Add (or subtract) the coefficients (number parts) of the like terms. The variable part stays the same.

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Combining Like Terms EXAMPLE Combining Like Terms Combine like terms. (a) 8a + 5a – a = 12a 8a + 5a – a No coefficient; write understood 1. 8a + 5a – 1a Change subtraction to addition of opposite. 8a + 5a + –1a 8a + 5a + –1a Find like terms. Use the distributive property. ( –1 ) a Add the coefficients. 12 a The variable part, a, stays the same.

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Combining Like Terms EXAMPLE Combining Like Terms Combine like terms. ALTERNATIVE METHOD (a) 8a + 5a – a = 12a 8a + 5a – a No coefficient; write understood 1. 8a + 5a – 1a Find like terms. Use the distributive property. ( – 1 ) a Add and subtract the coefficients. 12 a The variable part, a, stays the same.

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**–3x2 – 4x2 –3x2 + –4x2 –7 x2 Combining Like Terms**

EXAMPLE Combining Like Terms Combine like terms. (b) –3x2 – 4x2 = –7x2 –3x2 – 4x2 Change subtraction to addition of opposite. –3x2 + –4x2 Find like terms. Use the distributive property. ( – –4 ) x2 Add and subtract the coefficients. –7 x2 The variable part, x2, stays the same.

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**–3x2 – 4x2 –7 x2 Combining Like Terms EXAMPLE 2 Combining Like Terms**

Combine like terms. ALTERNATIVE METHOD (b) –3x2 – 4x2 = –7x2 –3x2 – 4x2 Find like terms. Use the distributive property. ( – – 4 ) x2 Subtract the coefficients. –7 x2 The variable part, x2, stays the same.

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**Simplifying Expressions**

EXAMPLE Simplifying Expressions Simplify each expression by combining like terms. (a) 2ac + 8c + 7ac = 9ac + 8c 2ac + 8c + 7ac Rewrite expression using commutative property. 2ac + 7ac + 8c Find like terms. Use the distributive property. ( ) ac + 8c Add inside parentheses. 9ac + 8c The expression is simplified.

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**Simplifying Expressions**

EXAMPLE Simplifying Expressions Simplify each expression by combining like terms. (b) 3n – 4 – n + 9 = 2n + 5 3n – – n + 9 Write 1 as the coefficient of n. 3n – – 1n + 9 Change subtraction to adding the opposite. 3n + – –1n + 9 Rewrite with like terms next to each other. 3n + –1n + – Use the distributive property. ( –1 )n + – Combine like terms. 2n + 5 The expression is simplified.

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**Simplifying Expressions**

EXAMPLE Simplifying Expressions Simplify each expression by combining like terms. (b) 3n – 4 – n + 9 = 2n + 5 ALTERNATIVE METHOD 3n – 4 – n + 9 Write 1 as the coefficient of n. 3n – 4 – 1n + 9 Rewrite with like terms next to each other. 3n – 1n – Combine like terms. 2n + 5 The expression is simplified.

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**Note on the Order of Listing Terms**

When combining like terms, we typically write the variable terms in alphabetical order. A constant term (number only) will be written last. So, in Examples 3(a) and 3(b), the preferred and alternative ways of writing the expressions are as follows: The simplified expression is 9ac + 8c (alphabetical order). However, by the commutative property of addition, 8c + 9ac is also correct. The simplified expression is 2n (constant written last). However, by the commutative property of addition, n is also correct.

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**Simplifying Multiplication Expressions**

EXAMPLE Simplifying Multiplication Expressions Simplify. (a) 4 ( –9x ) = –36x 4 • –9 • x ( ) Rewrite expression using associative property. 4 • –9 • x ( ) Multiply. –36 x The expression is simplified.

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**Simplifying Multiplication Expressions**

EXAMPLE Simplifying Multiplication Expressions Simplify. (b) –3 ( –2n ) = 6n –3 • –2 • n ( ) Rewrite expression using associative property. –3 • –2 • n ( ) Multiply. 6 n The expression is simplified.

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**Simplifying Multiplication Expressions**

EXAMPLE Simplifying Multiplication Expressions Simplify. (c) –5 ( 8y2 ) = –40y2 –5 • • y2 ( ) Rewrite expression using associative property. –5 • • y2 ( ) Multiply. –40 y2 The expression is simplified.

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**The Distributive Property**

Multiplication distributes over addition and subtraction as follows: 2 ( x ) can be written as 2 • x • 7 2x So, 2 ( x ) simplifies to 2x + 14. Stays as addition 6 ( x – 3 ) can be written as 6 • x – 6 • 3 6x – 18 So, 6 ( x – 3 ) simplifies to 6x – 18. Stays as subtraction

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**Using the Distributive Property**

EXAMPLE Using the Distributive Property Simplify. (a) 5 ( 2a – 3 ) can be written as 5 • 2a – 5 • 3 10a – 15 Stays as subtraction So, 5 ( 2a – 3 ) simplifies to 10a – 15.

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**Using the Distributive Property**

EXAMPLE Using the Distributive Property Simplify. (b) 8 ( 4n ) can be written as 8 • 4n • 7 32n Stays as addition So, 8 ( 4n ) simplifies to 32n + 56.

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**Using the Distributive Property**

EXAMPLE Using the Distributive Property Simplify. (c) –3 ( 9k ) can be written as –3 • 9k + –3 • 2 –27k –6 Stays as addition So, –3 ( 9k ) simplifies to –27k + –6. Using the definition of subtraction “in reverse”, we rewrite –27k + – as –27k – 6.

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**Using the Distributive Property**

EXAMPLE Using the Distributive Property ALTERNATIVE METHOD Simplify. (c) –3 ( 9k ) = –27k – 6 A negative ( –3 ) x a “positive” ( + 2 ) = “negative” 6 ( – 6 ). When you distribute, treat the operation within parentheses as the sign of the second term. In this example, as we distribute the –3 to the 2, we read it as “ –3 times positive 2”. –3 • 9k –3 ( 9k ) = –27k – 6 –3 • + 2

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**Simplifying a More Complex Expression**

EXAMPLE Simplifying a More Complex Expression Simplify: ( x – 5 ) = 4x – 13 ( x – 5 ) Do not add Use the distributive property. • x – 4 • 5 Do the multiplication. x – 20 Rewrite so that like terms are next to each other. 4x – 20 Subtract 7 – 20. 4x –13 Rewrite using the definition of subtraction “in reverse”. 4x – 13

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**Simplifying Expressions**

Chapter 2 Section 2 – Completed Written by John T. Wallace

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Columbus State Community College

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