1. Distributive Property, Combining Like Terms (1-4) Objective: Use the Distributive Property to evaluate and simplify expressions. Combine like terms.

2. Distributive Property The distributive property is used to simplify the product of a term and a sum or difference.. For any numbers a, b, and c:  a(b + c) = ab + ac  (b + c)a = ab + ac  a(b – c) = ab – ac  (b – c)a = ab – ac 3(2 + 5) = 3 2 + 3 5 = 6 + 15 = 21 4(9 – 7) = 4 9 – 4 7 = 36 – 28 = 8

3. Distributive Property The Symmetric Property allows the Distributive Property to be written as follows. –If a(b + c) = ab + ac, the ab + ac = a(b + c).

4. Example 1 Julio walks 5 days a week. He walks at a fast rate for 7 minutes and cools down for 2 minutes. Use the Distributive Property to write and evaluate an expression that determines the total number of minutes Julio walks. 5(7 + 2) = 5(7) + 5(2) = 35 + 10 = 45 minutes

5. Check Your Progress Choose the best answer for the following. –Susanne walks to school and home from school 5 days each week. She walks to school in 15 minutes and then walks home in 10 minutes. Rewrite 5(15 + 10) using the Distributive Property. Then evaluate to find the total number of minutes Susanne spends walking to and home from school. A.15 + 5 10; 65 minutes B.5 15 + 10; 85 minutes C.5 15 + 5 10; 125 minutes D.15 + 10; 25 minutes 5(15 + 10)= 5(15) + 5(10) = 75 + 50

6. Example 2 You can use the Distributive Property to make mental math easier. Use the Distributive Property to rewrite 12 82. Then evaluate. 12 82 = (10 + 2)82 = 82(10) + 82(2) = 820 + 164 = 984

7. Check Your Progress Choose the best answer for the following. –Use the Distributive Property to rewrite 6 54. Then evaluate. A.6(50); 300 B.6(50 4); 1200 C.6(50 + 4); 324 D.6(50 + 4); 654 6 54 = 6(50 + 4) = 6(50) + 6(4) = 300+ 24 = 324

8. Simplify Expressions The Distributive Property can be used to simplify algebraic expressions using variables. Like terms are terms that contain the same variables, with corresponding variables having the same power.

9. Simplify Expressions An expression is in simplest form when it contains no like terms or parenthesis. 5x 2 + 2x – 4 6a 2 + a 2 + 2a Three terms. No like terms. Three terms. Two like terms. Like Terms.

10. Example 3 Rewrite each expression using the Distributive Property. Then simplify. 12(y + 3) = 12(y) + 12(3) = 12y + 36 (y 2 + 8y – 2)4 = 4(y 2 ) + 4(8y) – 4(2) = 4y 2 + 32y – 8 -2(5x – 8) = (-2)(5x) – (-2)(8) = -10x – (-16) = -10x +16

11. Check Your Progress Choose the best answer for the following. –Simplify 6(x – 4). A.6x – 4 B.6x – 24 C.x – 24 D.6x + 2 6(x – 4) = 6(x) – 6(4)

12. Check Your Progress Choose the best answer for the following. –Simplify 3(x 3 + 2x 2 – 5x + 7). A.3x 3 + 2x 2 – 5x + 7 B.4x 3 + 5x 2 – 2x + 10 C.3x 3 + 6x 2 – 15x + 21 D.x 3 + 2x 2 – 5x + 21 3(x 3 + 2x 2 – 5x + 7) = 3(x 3 ) + 3(2x 2 ) – 3(5x) + 3(7)

13. Simplify Expressions The Distributive Property and the properties of equality can be used to show that 4k + 8k = 12k. In this expression, 4k and 8k are like terms.  4k + 8k = (4 + 8)k = 12k

14. Example 4 Simplify the following. Simplify 17a + 21a. 17a + 21a = (17 + 21)a = 38a Simplify 12b 2 – 8b 2 + 6b. 12b 2 – 8b 2 + 6b = (12 – 8)b 2 + 6b = 4b 2 + 6b

15. Check Your Progress Choose the best answer for the following. –Simplify 14x – 9x. A.5x 2 B.23x C.5 D.5x 14x – 9x = (14 – 9)x

16. Check Your Progress Choose the best answer for the following. –Simplify 6n 2 + 7n + 8n. A.6n 2 + 15n B.21n 2 C.6n 2 + 56n D.62n 2 6n 2 + 7n + 8n = 6n 2 + (7 + 8)n

17. Example 5 Use the expression six times the sum of x and y increased by four times the difference of 5x and y. a.Write an algebraic expression for the verbal expression. 6(x + y) + 4(5x – y) b.Simplify the expression and indicate the properties used. 6(x) + 6(y) + 4(5x) – 4(y) 6x + 6y + 20x – 4y 6x + 20x + 6y – 4y (6 + 20)x + (6 – 4)y 26x + 2y Distributive Property Multiply Commutative Property (Add.) Distributive Property Substitution

18. Check Your Progress Choose the best answer for the following. Use the expression three times the difference of 2x and y increased by two times the sum of 4x and y. –Write an algebraic expression for the verbal expression. A.3(2x + y) + 2(4x – y) B.3(2x – y) + 2(4x + y) C.2(2x – y) + 3(4x + y) D.3(x – 2y) + 2(4x + y)

19. Check Your Progress Choose the best answer for the following. Use the expression three times the difference of 2x and y increased by two times the sum of 4x and y. –Simplify the expression from Part A. A.2x + 4y B.11x C.14x – y D.12x + y 3(2x) – 3(y) + 2(4x) + 2(y) = 6x – 3y + 8x + 2y = 6x + 8x – 3y + 2y = (6 + 8)x + (-3 + 2)y

20. Coefficient The coefficient of a term is the numerical factor. For example, in 6ab, the coefficient is 6. In the coefficient is. In the term y, the coefficient is 1.