The Distributive Property and Simplifying Expressions Sections 2.5 – 2.8

Recall the distributive property of multiplication over addition... symbolically: a × (b + c) = a × b + a × c and pictorially (rectangular array area model): a × ba × ca bc

An example: 6 x 13 using your mental math skills... symbolically: 6 × (10 + 3) = 6 × × 3 and pictorially (rectangular array area model): 6 × 106 ×

Example 1-1a Use the Distributive Property to write as an equivalent expression. Then evaluate the expression. Answer: 52 Multiply. Add.

Example 1-1b Use the Distributive Property to write as an equivalent expression. Then evaluate the expression. ***It doesn’t matter which side of the parenthesis the number is on. The property works the same. Answer: 30 Multiply. Add.

Example 1-3a Use the Distributive Property to write as an equivalent algebraic expression. Simplify. Answer:

Example 1-3b Use the Distributive Property to write as an equivalent algebraic expression. Simplify. Answer:

Use the Distributive Property to write each expression as an equivalent algebraic expression. a. b. Example 1-3c Answer:

Example 1-4a Use the Distributive Property to write as an equivalent algebraic expression. Rewriteas Distributive Property Simplify. Definition of subtraction Answer:

Example 1-4b Use the Distributive Property to write as an equivalent algebraic expression. Distributive Property Simplify. Answer: Rewriteas

Algebraic terms and expressions To simplify algebraic expressions, we need to know certain “key words”. A letter that stands for a numerical value : VARIABLE A numerical value : CONSTANT A combination of variables, constants, and operations: EXPRESSION A number in front of a expression product : COEFFICIENT Expressions separated by addition or subtraction: TERMS Terms or expressions with the same variables and powers: LIKE TERMS

Simplify algebraic expressions, combine like terms Simplify means to combine as many terms as possible in an expression… Be sure to combine only like terms by addition or subtraction of the coefficients (same variables with identical exponents) Example: Simplify 4m 2 + 3m – 2m 2 + 5m. subtract coefficients of m with power 2 (4 - 2) (m 2 ) add coefficients of m with power 1 (3 + 5) (m) 2m 2 + 8m Example: Simplify ( s – 3t) + (2t – 3s). stack or line up coefficients of s and coefficients of t s – 3t -3s + 2t add coefficients of s and t vertically -2s + -t

GOAL 1 DIVIDING REAL NUMBERS 2.7 Division of Real Numbers VOCABULARY Reciprocal aka “Multiplicative Inverse” Since we use reciprocals when dividing, you must be able to write the reciprocal of a number. Remember: The reciprocal of the reciprocal of Since 0 does not have a reciprocal, we cannot divide by 0.

To divide a number a by a nonzero number b, multiply a by the multiplicative invers or reciprocal of b. DIVISION RULE EXAMPLE 1

Extra Example 1 Find each quotient. a. b. c.d.

GOAL 2 WORKING WITH ALGEBRAIC EXPRESSIONS 2.7 Division of Real Numbers EXAMPLE 2 Since we divide by multiplying by the reciprocal, the same rules about signs apply. Rules for dividing two real numbers: If the signs are the same, the product is _______. If the signs are different, the product is ________. positive negative

Extra Example 2 EXAMPLE 3 Simplify the expression: Multiply by the reciprocal. Use the distributive property. Simplify.

Extra Example 3 Evaluate the expression when x = –5 and y = –1. a.b.c. Undefined

Checkpoint 1. Simplify the expression –5d Evaluate the expression when s = –3 and t = –1. 0 EXAMPLE 4

Extra Example 4 A mountain climber descends 300 feet in 50 minutes. What is her velocity? EXAMPLE 5 VERBAL MODEL LABELS ALGEBRAIC MODEL SOLVE =Velocity Displacement Time v 50 min –300 feet