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Ch 2 Sec 2: Slide #1 Columbus State Community College Chapter 2 Section 2 Simplifying Expressions

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Ch 2 Sec 2: Slide #2 Simplifying Expressions 1.Combine like terms, using the distributive property. 2.Simplify expressions. 3.Use the distributive property to multiply.

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Ch 2 Sec 2: Slide #3 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. Three terms Separates the terms x 4y4y 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. 1x + 4y – 5

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Ch 2 Sec 2: Slide #4 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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Ch 2 Sec 2: Slide #5 Examples of Like Terms Like Terms 2x and – 4x1. 2. 3. 4. Variable parts match; both are x. 8b 7 and 2b 7 Variable parts match; both are b 7. – 6a 3 c 5 and 2a 3 c 5 Variable parts match; both are a 3 c 5. 4 and – 5No variable parts; numbers are like terms.

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Ch 2 Sec 2: Slide #6 Examples of Unlike Terms Unlike Terms 4x and – 2x 2 1. 2. 3. 4. Variable parts do not match; exponents are different. 3a and 3b Variable parts do not match; letters are different. m 2 n 3 and m 2 n 5 Variable parts do not match; exponents are different. 7v and – 5 Variable parts do not match; one term has a variable part, but the other term does not.

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Ch 2 Sec 2: Slide #7 Identifying Like Terms and Their Coefficients EXAMPLE 1 Identifying Like Terms and Their Coefficients (a)x + – 7x 2 + – 4xy + – 5x – 8 List the like terms in each expression. Then identify the coefficients of the like terms. 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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Ch 2 Sec 2: Slide #8 Identifying Like Terms and Their Coefficients EXAMPLE 1 Identifying Like Terms and Their Coefficients (b)3a 2 b + 2ab 2 – 4a 2 b + 8a 2 b 2 – a 2 b List the like terms in each expression. Then identify the coefficients of the like terms. The like terms are 3a 2 b, 4a 2 b, and a 2 b. The coefficient of 3a 2 b is 3, the coefficient of 4a 2 b is 4 (or – 4), and the coefficient of a 2 b is 1 (or – 1).

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Ch 2 Sec 2: Slide #9 Identifying Like Terms and Their Coefficients EXAMPLE 1 Identifying Like Terms and Their Coefficients (c)8m + 2 + 9n + 6mn + 7mn 2 – 5 List the like terms in each expression. Then identify the coefficients of the like terms. The like terms are 2 and 5 (or – 5). The like terms are constants (there are no variable parts).

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Ch 2 Sec 2: Slide #10 CAUTION Variable part is unchanged. Do not change x to x 2. CAUTION Notice that 4x and 7x is simplified to 11x, not to 11x 2.

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Ch 2 Sec 2: Slide #11 Combining Like Terms Step 1If there are any variable terms with no coefficient, write in the understood 1. Step 2If 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 3Find like terms (the variable parts match). Step 4Add (or subtract) the coefficients (number parts) of the like terms. The variable part stays the same.

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Ch 2 Sec 2: Slide #12 Combining Like Terms EXAMPLE 2 Combining Like Terms (a)8a + 5a – a Combine like terms. 8a + 5a – a 8a + 5a – 1a 8a + 5a + – 1a 12 a No coefficient; write understood 1. Change subtraction to addition of opposite. Find like terms. Use the distributive property. Add the coefficients. The variable part, a, stays the same. 8a + 5a + – 1a = 12a ( 8 + 5 + – 1 ) a

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Ch 2 Sec 2: Slide #13 Combining Like Terms EXAMPLE 2 Combining Like Terms (a)8a + 5a – a Combine like terms. 8a + 5a – a 8a + 5a – 1a 12 a No coefficient; write understood 1. Find like terms. Use the distributive property. Add and subtract the coefficients. The variable part, a, stays the same. = 12a ALTERNATIVE METHOD ( 8 + 5 – 1 ) a

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Ch 2 Sec 2: Slide #14 Combining Like Terms EXAMPLE 2 Combining Like Terms (b) – 3x 2 – 4x 2 Combine like terms. – 3x 2 – 4x 2 – 3x 2 + – 4x 2 – 7 x 2 Change subtraction to addition of opposite. Find like terms. Use the distributive property. Add and subtract the coefficients. The variable part, x 2, stays the same. = – 7x 2 ( – 3 + – 4 ) x 2

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Ch 2 Sec 2: Slide #15 Combining Like Terms EXAMPLE 2 Combining Like Terms (b) – 3x 2 – 4x 2 Combine like terms. – 3x 2 – 4x 2 ( – 3 – 4 ) x 2 Find like terms. Use the distributive property. Subtract the coefficients. The variable part, x 2, stays the same. = – 7x 2 – 7 x 2 ALTERNATIVE METHOD

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Ch 2 Sec 2: Slide #16 Simplifying Expressions EXAMPLE 3 Simplifying Expressions (a)2ac + 8c + 7ac Simplify each expression by combining like terms. 2ac + 8c + 7ac 2ac + 7ac + 8c ( 2 + 7 ) ac + 8c Rewrite expression using commutative property. Find like terms. Use the distributive property. Add inside parentheses. The expression is simplified. = 9ac + 8c 9ac + 8c

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Ch 2 Sec 2: Slide #17 Simplifying Expressions EXAMPLE 3 Simplifying Expressions (b)3n – 4 – n + 9 Simplify each expression by combining like terms. 3n – 4 – n + 9 3n – 4 – 1n + 9 3n + – 4 + – 1n + 9 Write 1 as the coefficient of n. Change subtraction to adding the opposite. Rewrite with like terms next to each other. Use the distributive property. = 2n + 5 3n + – 1n + – 4 + 9 Combine like terms.( 3 + – 1 )n + – 4 + 9 The expression is simplified.2n + 5

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Ch 2 Sec 2: Slide #18 Simplifying Expressions EXAMPLE 3 Simplifying Expressions (b)3n – 4 – n + 9 Simplify each expression by combining like terms. 3n – 4 – n + 9 3n – 4 – 1n + 9 3n – 1n – 4 + 9 Write 1 as the coefficient of n. Rewrite with like terms next to each other. Combine like terms. The expression is simplified. = 2n + 5 2n + 5 ALTERNATIVE METHOD

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Ch 2 Sec 2: Slide #19 Note on the Order of Listing Terms NOTE 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 + 5 (constant written last). However, by the commutative property of addition, 5 + 2n is also correct.

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Ch 2 Sec 2: Slide #20 Simplifying Multiplication Expressions EXAMPLE 4 Simplifying Multiplication Expressions (a)4 ( – 9x ) Simplify. 4 – 9 x – 36 x Rewrite expression using associative property. Multiply. The expression is simplified. = – 36x ( )

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Ch 2 Sec 2: Slide #21 Simplifying Multiplication Expressions EXAMPLE 4 Simplifying Multiplication Expressions (b) – 3 ( – 2n ) Simplify. – 3 – 2 n 6 n6 n Rewrite expression using associative property. Multiply. The expression is simplified. = 6n ( )

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Ch 2 Sec 2: Slide #22 Simplifying Multiplication Expressions EXAMPLE 4 Simplifying Multiplication Expressions (c) – 5 ( 8y 2 ) Simplify. – 5 8 y 2 – 40 y 2 Rewrite expression using associative property. Multiply. The expression is simplified. = – 40y 2 ( )

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Ch 2 Sec 2: Slide #23 The Distributive Property Multiplication distributes over addition and subtraction as follows: 2 ( x + 7 )can be written as 2 x + 2 7 Stays as addition 2x + 14 So, 2 ( x + 7 ) simplifies to 2x + 14. 6 ( x – 3 )can be written as 6 x – 6 3 Stays as subtraction 6x – 18 So, 6 ( x – 3 ) simplifies to 6x – 18.

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Ch 2 Sec 2: Slide #24 Using the Distributive Property EXAMPLE 5 Using the Distributive Property (a)5 ( 2a – 3 ) Simplify. can be written as 5 2a – 5 3 Stays as subtraction 10a – 15 So, 5 ( 2a – 3 ) simplifies to 10a – 15.

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Ch 2 Sec 2: Slide #25 Using the Distributive Property EXAMPLE 5 Using the Distributive Property (b)8 ( 4n + 7 ) Simplify. can be written as 8 4n + 8 7 Stays as addition 32n + 56 So, 8 ( 4n + 7 ) simplifies to 32n + 56.

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Ch 2 Sec 2: Slide #26 Using the Distributive Property EXAMPLE 5 Using the Distributive Property (c) – 3 ( 9k + 2 ) Simplify. can be written as – 3 9k + – 3 2 Stays as addition – 27k + – 6 So, – 3 ( 9k + 2 ) simplifies to – 27k + – 6. Using the definition of subtraction in reverse, we rewrite – 27k + – 6 as – 27k – 6.

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Ch 2 Sec 2: Slide #27 Using the Distributive Property EXAMPLE 5 Using the Distributive Property (c) – 3 ( 9k + 2 ) Simplify. –3 9k–3 9kWhen 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. = – 27k ALTERNATIVE METHOD – 3 + 2 – 6 – 3 ( 9k + 2 ) A negative ( – 3 ) x a positive ( + 2 ) = negative 6 ( – 6 ). = – 27k – 6

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Ch 2 Sec 2: Slide #28 Simplifying a More Complex Expression EXAMPLE 6 Simplifying a More Complex Expression 7 + 4 ( x – 5 ) Simplify:7 + 4 ( x – 5 ) Do not add 7 + 4. Use the distributive property. 7 + 4 x – 4 5Do the multiplication. 7 + 4x – 20 4x + 7 – 20 4x + – 13 4x – 13 Rewrite so that like terms are next to each other. Subtract 7 – 20. Rewrite using the definition of subtraction in reverse. = 4x – 13

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Ch 2 Sec 2: Slide #29 Simplifying Expressions Chapter 2 Section 2 – Completed Written by John T. Wallace

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