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Factoring Trinomials Module VII, Lesson 5 Online Algebra VHS@PWCS 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! Factors Multiply the following binomials. 1. (x + 5)(x + 7) x2 + 12x + 35 2. (2x – 3)(x + 4) 2x2 +5x -12 3. (x – 2)(3x – 1) 3x2 -7x + 1 (x + 5) and (x + 7) are factors of x2 + 12x + 7. (2x – 3) and (x + 4) are factors of 2x2 + 5x – 12. (x – 2) and (3x – 1) are factors of 3x2 – 7x + 1 When we multiply the binomials above we get a quadratic trinomial. Factoring Trinomials Factoring trinomials of the form x2 + bx + c,. Remember that if there is no coefficient in front of the x it is 1. Examples of this type of trinomial are: x2 + 4x - 3 x2 – 7x + 9 x2 + 6x + 26 Factoring Trinomials To factor x2 + 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. Factoring Trinomials Factor x2 + 5x + 6 1. 2. 3. Find the factors of c Find the sum of each pair of factors. 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) You can use FOIL to check: (x + 2)(x + 3) = x2 + 3x + 2x + 6 = x2 + 5x + 6 Factoring trinomials 1. 2. 3. 4. y2 – 4y – 45 Find the factors of -45. In this case since the c is negative we need one negative factor and one positive factor. -1 45 45 = 44 1 1+and -45 -45 = -44 -1+and -3 15 15 = 12 -3+and 3 3+and -15 -15 = -12 -5 + 9 = 4 -5 and 9 5 and -9 5 + -9 = -4 Find the sum of the factors. Use the factors that add up to -4. (y – 9)(y + 5) You can use foil to check. (y – 9)(y + 5) = y2 + 5y – 9y – 45 = y2 – 4y - 45 Try these on your own! 1. c2 – 2c + 1 1. (c – 1)(c – 1) 2. r2 + 6r – 16 2. (r – 2)(r + 8) 3. x2 + 10x + 25 3. (x + 5)(x + 5) Do you notice any patterns? If b and c are positive, then the factors you will use are both positive. x2 + 10x + 25 = (x + 5)(x + 5) If b is negative and c is positive, then the factors you will use are both negative. x2 – 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. x2 - 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. x2 + 5x – 6 = (x + 6)(x – 1) Factoring Trinomials – Factor by Grouping To factor trinomials of the form ax2 + 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: ax2 + (one of the factors)x + (the other factor)x + c. 5. 6. Factor the first 2 terms, then the second 2 terms. The goal when factoring is to get the same binomial. 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. Factor: 5x2 – 2x - 7 5 x -7 = -35 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. 5x2 + 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 5x2 + 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) 1. 2. 1. 2. 3. 4. Find the product (multiply) of a and c. Find the factors of the product of a and c The patterns that we found still apply. Use the factors that add up to b. Write the quadratic as: ax2 + (one of the factors)x + (the other factor)x + c. 5. 6. Factor the first 2 terms, then the second 2 terms. The goal when factoring is to get the same binomial. Write as: (same binomial)(GCF of First pair + GCF of second pair) Factor 3x2 + 13x - 10 1. 2. 3. 4. Find the product (multiply) of a and c. Find the factors of the product of a and c The patterns that we found still apply. Use the factors that add up to b. Write the quadratic as: ax2 + (one of the factors)x + (the other factor)x + c. 5. 6. Factor the first 2 terms, then the second 2 terms. The goal when factoring is to get the same binomial. Write as: (same binomial)(GCF of First pair + GCF of second pair) 1. 3 x -10 = -30 2. -1 and 30 3. 4. 5. 6. -2 and 15 -3 and 10 -5 and 6 -2 + 15 = 13 3x2 + -2x + 15x – 10 x(3x – 2) + 5(3x – 2) (x + 5)(3x – 2) Factor: 3x2 + 17x + 20 1. 2. 3. 4. Find the product (multiply) of a and c. Find the factors of the product of a and c The patterns that we found still apply. Use the factors that add up to b. Write the quadratic as: 1. 3 x 20 = 60 2. 1 and 60 ax2 + (one of the factors)x + (the other factor)x + c. 5. 6. Factor the first 2 terms, then the second 2 terms. The goal when factoring is to get the same binomial. Write as: (same binomial)(GCF of First pair + GCF of second pair) 3. 4. 5. 6. 2 and 30 3 and 20 4 and 15 5 and 12 6 and 10 5 + 12 = 17 3x2 + 12x + 5x + 20 3x(x + 4) + 5(x + 4) (3x + 5)(x + 4) Try these on your own. 1. 2w2 – w – 3 2. 2t2 + 3t – 2 3. 6x2 + 10x + 4 1. 2w2 + 2w – 3w – 3 2w(w + 1) – 3(w + 1) (2w – 3)(w + 1) 2. 2t2 – 1t + 4t – 2 t(2t – 1) + 2(2t – 1) (t + 2)(2t – 1) 3. 6x2 + 6x + 4x + 4 6x(x + 1) + 4(x + 1) (6x + 4)(x + 1) 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!