Warm Up 8/13/09 Simplify. 1. 2. 4.4 – 32 +6.1 4. 3(8 + 3) 2. 4.4 – 32 +6.1 4. 3(8 + 3) 3. (3-2)(2-7) + 1 Evaluate Each Expression. 5. 2 + (x + 2)2 for x = 3 6.

Objectives Use the Commutative, Associative, and Distributive Properties to simplify expressions. Combine like terms.

Vocabulary term like terms coefficient

The Commutative and Associative Properties of Addition and Multiplication allow you to rearrange an expression to simplify it.

The Distributive Property is used with Addition to Simplify Expressions. The Distributive Property also works with subtraction because subtraction is the same as adding the opposite.

Directions: Write the product using the Distributive Property Directions: Write the product using the Distributive Property. Then simplify.

Example 1 3(x – 4) 3(x) – 3(4) Multiply the 3 times the x and the 4 Simplify each term 3x - 12

Example 2 -5(2x + 7) Multiply -5 times 3x and 7 -5(2x) + (-5)(7) Simplify each term. -10x - 35

Example 3 6(-7x + 8) 6(-7x) + 6(8) -42x + 48

Example 4 -5(3x2 + 7) -5(3x2 + (-5)(7) -15x2 -35

Like Terms What are “like terms”

The terms of an expression are the parts to be added or subtracted The terms of an expression are the parts to be added or subtracted. Like terms are terms that contain the same variables raised to the same powers. Constants are also like terms. Like terms Constant 4x – 3x + 2

A coefficient is a number multiplied by a variable A coefficient is a number multiplied by a variable. Like terms can have different coefficients. A variable written without a coefficient has a coefficient of 1. Coefficients 1x2 + 3x

Using the Distributive Property can help you combine like terms Using the Distributive Property can help you combine like terms. You can factor out the common factors to simplify the expression. 7x2 – 4x2 = (7 – 4)x2 Factor out x2 from both terms. = (3)x2 Perform operations in parenthesis. = 3x2 Notice that you can combine like terms by adding or subtracting the coefficients and keeping the variables and exponents the same.

Directions: Simplify the expression by combining like terms. Write each example and solution.

Example 5 72p – 25p 72p – 25p 72p and 25p are like terms. 47p Subtract the coefficients.

Example 6 A variable without a coefficient has a coefficient of 1. and are like terms. Write 1 as . Add the coefficients.

Example 7 0.5m + 2.5n 0.5m + 2.5n 0.5m and 2.5n are not like terms. 0.5m + 2.5n Do not combine the terms.

Example 8 a. 16p + 84p 16p + 84p 16p + 84p are like terms. 100p Add the coefficients. b. –20t – 8.5t2 –20t – 8.5t2 20t and 8.5t2 are not like terms. –20t – 8.5t2 Do not combine the terms. c. 3m2 + m3 3m2 + m3 3m2 and m3 are not like terms. 3m2 + m3 Do not combine the terms.

Simplify and Justify Distribute and Combine Like Terms Put it all together! Copy each example and solution.

Example 9 Simplify 14x + 4(2 + x). Justify each step. Procedure Justification 1. 14x + 4(2 + x) 2. 14x + 4(2) + 4(x) Distributive Property 3. 14x + 8 + 4x Multiply. Commutative Property 4. 14x + 4x + 8 5. (14x + 4x) + 8 Associative Property 6. 18x + 8 Combine like terms.

Example 10 Simplify 6(x – 4) + 9. Justify each step. Procedure Justification 1. 6(x – 4) + 9 2. 6(x) – 6(4) + 9 Distributive Property 3. 6x – 24 + 9 Multiply. Combine like terms. 4. 6x – 15

Example 11 Simplify −12x – 5x + 3a + x. Justify each step. Procedure Justification 1. –12x – 5x + 3a + x 2. –12x – 5x + x + 3a Commutative Property 3. –16x + 3a Combine like terms.

Lesson Summary Simplify each expression by combining like terms. Justify each step with an operation or property. 1. 2. 14c2 – 9c 14c2 – 9c 3. 301x – x 300x 4. 24a + b2 + 3a + 2b2 27a + 3b2

Lesson Summary 5. 4(x + 5) 4x + 20 6. -5(3x – 7) -15x + 35 Simplify each expression using the distributive property. 5. 4(x + 5) 4x + 20 6. -5(3x – 7) -15x + 35 Simplify each expression by using the distributive property and combining like terms. 7. 4(x – 5) + 2x 6x - 20 8. -3(x + 2) – 4 -3x - 10