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Factoring Quadratic Trinomials …beyond the guess and test method. Topics 1. 2. 3. 4. 5. 6. 7. Standard Form When c is positive and b is positive When c is positive and b is negative When c is negative When the trinomial is not factorable When a does not equal 1 When there is a GCF Standard Form The standard form of any 2 ax bx c quadratic trinomial is So, in 3x 4x 1... 2 a=3 b=-4 c=1 Now you try. x 7x 2 2 a = ?? b = ?? c = ?? Click here when you are ready to check your answers! Recall The standard form of any 2 ax bx c quadratic trinomial is So, in 2x 2 x 5... a=2 b = -1 c=5 Try Another! 4x x 2 2 a = ?? b = ?? c = ?? Go on to factoring! Factoring when a=1 and c > 0. First list all the factors of c. x 2 8x 12 1 12 2 6 3 4 Find the pair that adds to ‘b’ 1 12 2 6 3 4 These numbers are used in the factored expression. x 2x 6 Now you try. 1. 2. 3. x 2 8x 15 x 2 10x 21 x 2 9x 20 Click here when you are ready to check your answers! Recall x 2 10x 24 We need to list the factors of c. So we get: x 4x 6 1 24 2 12 3 8 4 6 Try some others! x 6x 9 2 1. (x+3)(x+3) 2. (x+2)(x+3) 2. x 2 8x 7 x 7x 6 2 (x+1)(x+6) (x+2)(x+3) 2. x 2 8x 7 Go on to factoring where b is negative! Factoring when c >0 and b < 0. Since a negative number times a negative number produces a positive answer, we can use the same method. Just remember to use negatives in the expression! Let’s look at x 13x 12 2 First list the factors of 12 1 12 We need a sum of -13 2 6 3 4 Make sure both values are negative! x 12x 1 Now you try. 1. 2. 3. x 2 5x 4 x 2 9x 14 x 2 13x 42 Click here when you are ready to check your answers! Recall x 6x 8 2 In this case, one factor should be positive and the other negative. 1 8 2 4 We need a sum of -6 x 2x 4 Try some others! 1. x 7x 12 2 (x-3)(x-4) 2. (x-3)(x+4) 2. x 2 8x 7 x 4x 4 2 (x-2)(x-2) (x-1)(x-4) 2. x 2 8x 7 Go on to factoring where c is negative! Factoring when c < 0. We still look for the factors of c. However, in this case, one factor should be positive and the other negative. Remember that the only way we can multiply two numbers and come up with a negative answer, is if one is number is positive and the other is negative! Let’s look at x x 12 2 In this case, one factor should be positive and the 1 other negative. 2 We need a sum of -1 x 3x 4 3 12 6 4 Now you try. 1. 2. 3. 4. x 2 3x 4 x 2 x 20 x 2 4 x 21 x 2 10x 56 Click here when you are ready to check your answers! x 3x 18 Recall 2 In this case, one factor should be positive and the 1 other negative. 2 We need a sum of 3 x 3 18 12 6 3x 6 Try some others! 1. x 2x 15 2 (x-3)(x+5) 2. (x+3)(x-5) 2. x 2 8x 7 x x 30 2 (x-5)(x+6) (x-6)(x-5) 2. x 2 8x 7 Go on to trinomials that are not factorable Prime Trinomials Sometimes you will find a quadratic trinomial that is not factorable. You will know this when you cannot get b from the list of factors. When you encounter this write not factorable or prime. Here is an example… x 3x 18 2 1 18 2 9 3 6 Since none of the pairs adds to 3, this trinomial is prime. Now you try. x 2 6x 4 factorable prime x 2 10x 39 factorable prime x 2 5x 7 factorable prime Go on to factoring when a≠1 When a ≠ 1. Instead of finding the factors of c: Multiply a times c. Then find the factors of this product. 1 70 2 7x 19x 10 a c 70 2 35 5 14 7 10 We still determine the factors that add to b. So now we have x 5x 14 1 70 2 35 5 14 7 10 But we’re not finished yet…. Since we multiplied in the beginning, we need to divide in the end. Divide each constant by a. 5 14 x x 7 7 Simplify, if possible. 5 x x 2 7 Clear the fraction in each binomial factor 7x 5x 2 Recall 2x 3x 9 2 Multiply a times c. List factors. Write 2 binomials with the factors that add to b Divide each constant by a. Simplify, if possible. Clear the fractions in each factor 2 9 18 1 2 18 9 3 6 x 6x 3 6 3 x x 2 2 3 x 3 x 2 x 32x 3 Try some others! Now you try. 1. 4x 4x 3 2. 3x 5x 12 3. 6x 23x 7 2 2 2 Click here when you are ready to check your answers! 1. 2x 9x 5 2 (2x-1)(x+5) 2. (2x+5)(x+1) 2. x 2 8x 7 4x 6x 5 2 (2x-5)(2x+1) (4x+5)(x-1) 2. x 2 8x 7 Go on to trinomials that have a GCF Sometimes there is a GCF. If so, factor it out first. 2 15 30 1 2 3 5 30 15 10 6 Ex) 4x 2x 30 2 22x 2 x 15 2x 6x 5 6 5 2x x 2 2 5 2x 3x 2 2x 35x 2 Now you try. 1. 4 x 2 16x 12 2. 6x 2 10x 6 Click here when you are ready to check your answers! Recall 45x 35x 10 2 First factor out the GCF. Then factor the remaining trinomial. 9 times 2 = 18 1 2 3 18 9 6 5 9x 7x 2 2 5x 2x 9 2 9 5 x x 9 9 59x 2x 1 59x 2x 1 Try some others! 1. 6x 30x 36 2 6(x-1)(x+6) 2. 2. x 2 8x 7 4x 14x 10 2 2(2x+1)(x+5) (6x+6)(x-6) 2(2x+5)(x+1) 2. x 2 8x 7 Did you get these answers? a 1 b 7 c 2 Yes No Did you get these answers? 1. x 3x 5 2. x 3x 7 x 5x 4 3. Yes No Did you get these answers? 1. x 1x 4 2. x 2x 7 x 6x 7 3. Yes No Did you get these answers? 1. 2. 3. 4. x 1x 4 x 4 x 5 x 3x 7 x 4 x 14 Yes No Did you get these answers? 1. 2. 3. 2x 12x 3 x 33x 4 3x 12x 7 Yes No Did you get these answers? 1. 4x 1x 3 2. prime Yes No Good Job! You have completed Standard Form! Good Job! You have completed factoring “When c is positive and b is positive”! Good Job! You have completed factoring “When c is positive and b is negative”! Good Job! Good Job! You have completed factoring “When a does not equal 1”! Good Job! You have completed factoring “When c is negative”! Good Job! Good Job! Good Job! Good Job! Good Job! Good Job! You have completed factoring “When there is a GCF”! Review and Try Again! Review and Try Again! Review and Try Again! Review and Try Again! Review and Try Again! Try Again! Try Again! Try Again! Try Again! Try Again! Review and Try Again! Try Again! Review and Try Again!