Simplifying and Factoring Algebraic Expressions Overview

1. 7.EE.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

2. Objective This presentation is designed to give a brief review of simplifying and factoring algebraic expressions

3. Algebraic Expressions An algebraic expression is a collection of real numbers, variables, grouping symbols and operation symbols. Here are some examples of algebraic expressions.

4. Consider the example: The terms of the expression are separated by addition. There are 3 terms in this example and they are . The coefficientof a variable term is the real number factor. The first term has coefficient of 5. The second term has an unwritten coefficient of 1. The last term , -7, is called a constant since there is no variable in the term.

5. Let’s begin with a review of two important skills for simplifying expression, using the Distributive Property and combining like terms. Then we will use both skills in the same simplifying problem.

6. Distributive Property a ( b + c ) = ba + ca To simplify some expressions we may need to use the Distributive Property Do you remember it? Distributive Property

7. Examples Example 1: 6(x + 2) Distribute the 6. 6 (x + 2) = x(6) + 2(6) = 6x + 12 Example 2: -4(x – 3) Distribute the –4. -4 (x – 3) = x(-4) –3(-4) = -4x + 12

8. Practice Problem Try the Distributive Property on -7 ( x – 2 ) . Be sure to multiply each term by a –7. -7 ( x – 2 ) = x(-7) – 2(-7) = -7x + 14 Notice when a negative is distributed all the signs of the terms in the ( )’s change.

9. Example 3: (x – 2) = 1( x – 2 ) = x(1) – 2(1) = x - 2 Notice multiplying by a 1 does nothing to the expression in the ( )’s. Example 4: -(4x – 3) = -1(4x – 3) = 4x(-1) – 3(-1) = -4x + 3 Notice that multiplying by a –1 changes the signs of each term in the ( )’s. Examples with 1 and –1.

10. Like Terms Like terms are terms with the same variables raised to the same power. Hint: The idea is that the variable part of the terms must be identical for them to be like terms.

11. Like Terms 5x , -14x -6.7xy , 02xy The variable factors are identical. Unlike Terms 5x , 8y The variable factors are not identical. Examples

12. Combining Like Terms Recall the Distributive Property a (b + c) = b(a) +c(a) To see how like terms are combined use the Distributive Property in reverse. 5x + 7x = x (5 + 7) = x (12) = 12x

13. Example All that work is not necessary every time. Simply identify the like terms and add their coefficients. 4x + 7y – x + 5y = 4x – x + 7y +5y = 3x + 12y

14. Collecting Like Terms Example

15. Both Skills This example requires both the Distributive Property and combining like terms. 5(x – 2) –3(2x – 7) Distribute the 5 and the –3. x(5) - 2(5) + 2x(-3) - 7(-3) 5x – 10 – 6x + 21 Combine like terms. - x+11

16. Simplifying Example

17. Simplifying Example Distribute.

18. Simplifying Example Distribute.

19. Simplifying Example Distribute. Combine like terms.

20. Simplifying Example Distribute. Combine like terms.

21. Factoring Algebraic Expressions You can factor linear expressions by using the Greatest Common Factor, or GCF.

22. Remember, using the GCF means finding the largest factor of two or more numbers! Example: 24 & 32 Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 Factors of 32: 1, 2, 4, 8, 16, 32 GCF = 8

23. 4x+ 8 = 4(x) + 4(2) = 4(x+ 2) Using the distributive property in reverse: 4x + 8 Factors of 4: 1, 2, 4 Factors of 8: 1, 2, 4, 8 4x = 4(x) 8 = 4(2)

24. Another example: 24x - 32 24x = 8(3x) 32 = 8(4) 24x- 32 = 8(3x) - 8(4) = 8(3x- 4)

25. Now you try! 1. Factor: 9y + 15 3(3y + 5) 2. Factor: 24x - 60 12(2x + 5) 3. Factor: 12x - 18 6(2x – 3) -3(x + 6) 4. Factor: -3x - 18

26. Sample test questions:

27. Another one ~

28. Now, let’s practice! 