1. Factoring Trinomials Module VII, Lesson 5 Online Algebra VHS@PWCS

2. Factoring  Factors are integers that divide another integer evenly.  In a multiplication problem factors are the numbers that are being multiplied to get the product (the answer).  What are the factors of 90?  1 x 90  2 x 45  3 x 30  5 x 18  9 x 10 If we multiply these factors together the product is 90!

3. Factors Multiply the following binomials. 1. (x + 5)(x + 7) x 2 + 12x + 35 2. (2x – 3)(x + 4) 2x 2 +5x -12 3. (x – 2)(3x – 1) 3x 2 -7x + 1 (x + 5) and (x + 7) are factors of x 2 + 12x + 7. (2x – 3) and (x + 4) are factors of 2x 2 + 5x – 12. (x – 2) and (3x – 1) are factors of 3x 2 – 7x + 1 When we multiply the binomials above we get a quadratic trinomial.

4. Factoring Trinomials Factoring trinomials of the form x 2 + bx + c,. Remember that if there is no coefficient in front of the x it is 1. Examples of this type of trinomial are: x 2 + 4x - 3 x 2 – 7x + 9 x 2 + 6x + 26

5. Factoring Trinomials To factor x 2 + bx + c: 1. Find the factors of c 2. Find the sum of each pair of factors. 3. Use the factors of c that add up to b to put into binomials as follows: (x + one of the factors)(x + the other factor) This looks kind of confusing so lets try it with numbers.

6. Factoring Trinomials 1. Find the factors of c 2. Find the sum of each pair of factors. 3. Use the factors of c that add up to b to put into binomials as follows: (x + one of the factors)(x + the other factor)  Factors of 6 6 and 1 2 and 3  Sum of the factors 6 + 1 = 7 2 + 3 = 5  (x + 2)(x + 3) Factor x 2 + 5x + 6 You can use FOIL to check: (x + 2)(x + 3) = x 2 + 3x + 2x + 6 = x 2 + 5x + 6

7. Factoring trinomials y 2 – 4y – 45 1. Find the factors of -45. In this case since the c is negative we need one negative factor and one positive factor. -1 and 45 1 and -45 -3 and 153 and -15 -5 and 95 and -9 2. Find the sum of the factors. 3. Use the factors that add up to -4. (y – 9)(y + 5) 4. You can use foil to check. (y – 9)(y + 5) = y 2 + 5y – 9y – 45 = y 2 – 4y - 45 -1 + 45 = 44 -3 + 15 = 12 -5 + 9 = 4 1 + -45 = -44 3 + -15 = -12 5 + -9 = -4

8. Try these on your own! 1. c 2 – 2c + 1 2. r 2 + 6r – 16 3. x 2 + 10x + 25 1. (c – 1)(c – 1) 2. (r – 2)(r + 8) 3. (x + 5)(x + 5)

9. Do you notice any patterns?  If b and c are positive, then the factors you will use are both positive. x 2 + 10x + 25 = (x + 5)(x + 5)  If b is negative and c is positive, then the factors you will use are both negative. x 2 – 5x + 6 = (x -2)(x – 3)  If b is negative and c is negative, then one factor will be positive and the other will be negative. The negative number must have a larger absolute value. x 2 - 5x – 6 = (x – 6)(x + 1)  If b is negative and c is positive, then one factor will be positive and the other will be negative. The positive number must have a larger absolute value. x 2 + 5x – 6 = (x + 6)(x – 1)

10. Factoring Trinomials – Factor by Grouping To factor trinomials of the form ax 2 + bx + c, we use what we call factor by grouping. 1. Find the product (multiply) of a and c. 2. Find the factors of the product of a and c. 3. Use the factors that add up to b. 4. Write the quadratic as: ax 2 + (one of the factors)x + (the other factor)x + c. 5. Factor the first 2 terms, then the second 2 terms. The goal when factoring is to get the same binomial. 6. Write as: (same binomial)(GCF of First pair + GCF of second pair) All this is pretty difficult to explain so we will do quite a few examples.

11. Factor: 5x 2 – 2x - 7 1. Find the product (multiply) of a and c. 2. Find the factors of the product of a and c The patterns that we found still apply. 3. Use the factors that add up to b. 4. Write the quadratic as: ax 2 + (one of the factors)x + (the other factor)x + c. 5. Factor the first 2 terms, then the second 2 terms. The goal when factoring is to get the same binomial. 6. Write as: (same binomial)(GCF of First pair + GCF of second pair) 1. 5 x -7 = -35 2. Since the product is negative we need one positive and one negative, with the larger negative. 1 and – 35 5 and – 7 3. 5 + -7 = -2 4. 5x 2 + 5x + -7x – 7 Notice that the middle two terms add up to -2x, the middle term in the trinomial. 5x + -7x = -2x 5. 5x(x + 1) – 7(x + 1) 5x is the GCF of 5x 2 + 5x -7 is the GCF of -7x – 7 x + 1 is the binomial left for both when you pull out the GCF 6. (x + 1)(5x – 7)

12. Factor 3x 2 + 13x - 10 1. Find the product (multiply) of a and c. 2. Find the factors of the product of a and c The patterns that we found still apply. 3. Use the factors that add up to b. 4. Write the quadratic as: ax 2 + (one of the factors)x + (the other factor)x + c. 5. Factor the first 2 terms, then the second 2 terms. The goal when factoring is to get the same binomial. 6. Write as: (same binomial)(GCF of First pair + GCF of second pair) 1. 3 x -10 = -30 2. -1 and 30 -2 and 15 -3 and 10 -5 and 6 3. -2 + 15 = 13 4. 3x 2 + -2x + 15x – 10 5. x(3x – 2) + 5(3x – 2) 6. (x + 5)(3x – 2)

13. Factor: 3x 2 + 17x + 20 1. Find the product (multiply) of a and c. 2. Find the factors of the product of a and c The patterns that we found still apply. 3. Use the factors that add up to b. 4. Write the quadratic as: ax 2 + (one of the factors)x + (the other factor)x + c. 5. Factor the first 2 terms, then the second 2 terms. The goal when factoring is to get the same binomial. 6. Write as: (same binomial)(GCF of First pair + GCF of second pair) 1. 3 x 20 = 60 2. 1 and 60 2 and 30 3 and 20 4 and 15 5 and 12 6 and 10 3. 5 + 12 = 17 4. 3x 2 + 12x + 5x + 20 5. 3x(x + 4) + 5(x + 4) 6. (3x + 5)(x + 4)

14. Try these on your own. 1. 2w 2 – w – 3 2. 2t 2 + 3t – 2 3. 6x 2 + 10x + 4 1. 2w 2 + 2w – 3w – 3 2w(w + 1) – 3(w + 1) (2w – 3)(w + 1) 2. 2t 2 – 1t + 4t – 2 t(2t – 1) + 2(2t – 1) (t + 2)(2t – 1) 3. 6x 2 + 6x + 4x + 4 6x(x + 1) + 4(x + 1) (6x + 4)(x + 1)

15. Factoring Review Remember that factors are: Integers that divide another integer evenly In a multiplication problem they are the numbers that you multiply together The factors of a quadratic trinomial are 2 binomials Look for patterns!