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Columbus State Community College
Chapter 2 Section 2 Simplifying Expressions

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

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.

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.

–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.

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

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.

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).

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).

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.

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.

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.

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.

–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.

–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.

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.

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.

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.

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.

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.

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.

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.

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

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.

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.

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.

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

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

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