1. Bell Work: Simplify 1 + c w w 1 c

2. Answer: (1 + c)c w

3. Lesson 69: Factoring Trinomials

4. We say that a monomial is a single expression of the form ax, where a is any real number and n is any whole number. thus the following expressions are monomials: 4 6x -2x4.163x n 2 15 4

5. The number 4 can be classified as a monomial because it can be thought of as 4x. 0

6. A binomial is the indicated algebraic sum of two monomials and a trinomial is the indicated algebraic sum of three monomials. We use the word polynomial as the general descriptive term to describe monomials, binomials, trinomials and algebraic expressions that are the indicated sum of four or more monomials.

7. We are familiar with the vertical format for multiplying binomials, as shown here. x – 6 x + 3 x – 6x + 3x – 18 x – 3x – 18 2222

8. The product is a trinomial. We call these trinomials quadratic trinomials. The word quadratic tells us that the highest power of the variable is 2.

9. To reverse the process and factor the trinomials into a product of binomials, we must observe the pattern that developed when we did the multiplications. Note that:

10. 1) The first term of the trinomial is the product of the first terms of the binomials 2) The last term of the trinomial is the product of the last terms of the binomials 3) The coefficient of the middle term of the trinomial is the sum of the last terms of the binomials 4) If all signs in the trinomial are positive, all signs in both binomials are positive. If a negative sign appears in the trinomial, at least one of the terms of the binomials is negative

11. We use these observations to help us factor trinomials. To factor the trinomial x – 3x – 18 We first write down two sets of parentheses to form an indicated product. ( )( ) 2

12. Since the first term in the trinomial is the product of the first terms of the binomials, we enter x as the first term of each binomial. (x )(x )

13. Now the product of the last terms of the binomials must equal -18, their sum must equal -3, and at least one of them must be negative. There are six pairs of factors of -18.

14. (-18)(1) = -18(18)(-1) = -18 (2)(-9) = -18(-2)(9) = -18 (3)(-6) = -18(-3)(6) = -18

15. Their sums are: (-18) + (1) = -17(18) + (-1) = 17 (2) + (-9) = -7(-2) + (9) = 7 (3) + (-6) = -3(-3) + (6) = 3

16. Note that while all six pairs have a product of -18, only one pair (3 and -6) sums to -3. therefore, the last terms of the binomials are 3 and -6, and so (x + 3) and (x – 6) are the factors of x – 3x – 18 because (x + 3)(x – 6) = x – 3x – 18. 2 2

17. Thus, the general approach to factoring a quadratic trinomial that has a leading coefficient of 1 is to determine the pair of integral factors of the last term of the trinomial whose sum equals the coefficient of the middle term.

18. Example: Factor x – 8x + 16 2

19. Answer: (x – 4)(x – 4)

20. Example: Factor x – 14x – 15 2

21. Answer: (x – 15)(x + 1)

22. Practice: Factor x + 3x – 10 2

23. Answer: (x + 5)(x – 2)

24. Practice: Factor -5x + x + 6 2

25. Answer: (x – 3)(x – 2)

26. Practice: Factor x + 5 + 6x 2

27. Answer: (x + 5)(x + 1)

28. HW: Lesson 69 #1-30