Title: Factoring Using the Distributive Property

1. Warm up: p. 479 #72-86 even Title: Factoring Using the Distributive Property EQ: How do we factor polynomials by using the distributive property? How do we solve quadratic equations of the form ax + bx + c?

2. Factoring by using the distributive property • Express a polynomial as the product of a monomial factor and a polynomial factor. Example: Factored form:

3. Notes cont. • Factoring involves finding the GCF Example: • Now you write each term as the product of the GCF (divide it out) Example: • This will give you the factored form of:

4. Use the Distributive Property to factor . First, find the CGF of 15x and . Factor each number. GFC: Rewrite each term using the GCF. Simplify remaining factors. Distributive Property Example 2-1a Circle the common prime factors. Write each term as the product of the GCF and its remaining factors. Then use the Distributive Property to factor out the GCF.

5. Answer: The completely factored form of is Example 2-1a

6. Use the Distributive Property to factor . Factor each number. GFC: or Rewrite each term using the GCF. Distributive Property Answer: The factored form of is Example 2-1a Circle the common prime factors.

7. Use the Distributive Property to factor each polynomial. a. b. 6 Answer: Example 2-1b

8. Grouping to factor • If a polynomial has four or more terms it helps to group the polynomial and then factor. This means you take and split the polynomial into pairs. HINTS for grouping: • There are four or more terms • Terms with common factors should be grouped together. • The two common factors are identical or additive inverses of each other. Example:

9. Factor Group terms with common factors. Factor the GCFfrom each grouping. Answer: Distributive Property Example 2-2a

10. Factor Answer: Example 2-2b

11. The additive inverse property • Recognizing the polynomial as additive inverses can be VERY helpful when factoring by grouping. Additive inverses are like (-x-7) (7+x). You know they are additive inverses bc when you add them together the sum is 0. Parenthesis are identical except for signs! You need to pull out a negative in one of the outside numbers • Example:

12. Factor Group terms with common factors. Parenthesis are identical except for signs! You need to pull out a negative in one of the outside numbers Answer: Distributive Property Example 2-3a = -3a (-5 + b) + 4 (b – 5) What is outside goes in one parenthesis and what is inside goes onto another.

13. Factor Answer: Example 2-3b

14. Zero product property • If the product of two factors is 0, then at least one of the two factors is 0.

15. Solve an equation in factored form • Set up the two binomials so they are equal to zero and then solve for the variable. Example:

16. Solve Then check the solutions. If , then according to the Zero Product Property either or Original equation or Set each factor equal to zero. Solve each equation. Answer: The solution set is Example 2-4a

17. Check Substitute 2 and for x in the original equation. Example 2-4a

18. Solve Then check the solutions. Example 2-4b Answer: {3, –2}

19. Solve and equation by factoring • Write the equation so it is in the form of ab=0 • Then solve for x. Example:

20. Solve Then check the solutions. Write the equation so that it is of the form Original equation Subtract from each side. Factor the GCF of 4y andwhich is 4y. Zero Product Property or Solve each equation. Example 2-5a

21. Answer: The solution set is Check by substituting 0 and for y in the original equation. Example 2-5a

22. Solve Answer: Example 2-5b

23. Factoring • You always need to look for a GCF • If you are grouping you need to make sure you put like terms together before you put in ( ) That means that you may need to reorganize the polynomial • Always look for difference of perfect squares