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5.6 Factoring Trinomials ax bx c 2 How to Factor • 1. Write the trinomial in descending powers of one variable. • 2. Factor out any GCF (including (-1) to make the 1st term positive) • 3. Test the trinomial for factorability. 2 – Check the value of b 4ac • If it’s a perfect square, then it’s factorable into 2 different binomials • If it’s zero, then it’s factorable and the factors will be the same More steps • 4. When the sign of the 1st term of the trinomial is positive: – And the sign of the 3rd term is positive, the signs in each parentheses will be the same x 6x 5 x 5 x 1 2 – And the sign of the 3rd term is negative, the signs in each parentheses will be different x 6x 7 x 7 x 1 2 More steps continued • 5. Try different combos until you find the one that works. – If the leading coefficient is 1 • Pick the factorization where the sum of the factors of c is equal to the coefficient of the middle term (b) – If the leading coefficient is not 1 • Can use trial and error • Can use the key method Step 5 Cont’d (Key Method) • i. Find product of ac (this is the key #) • ii. Find 2 factors of the key # whose sum is b • iii. Use the factors of the key # as coefficient of 2 2 terms to be placed between ax and c • iv. Factor using grouping Example of key method 15a 17 a 4 2 i. ac 15 4 60 ii. factors of 60 that add up to 17 20 and 3 iii. 15a 3a 20a 4 2 iv. 3a 5a 1 4 5a 1 5a 1 3a 4 • 6. Check by multiplying out. Another example – key method 6 x 17 x 12 2 i. ac 6 12 72 ii. factors of 72 that add up to 17 (b term) 9 and 8 iii. 6 x 9 x 8x 12 2 iv. 3x 2x 3 4 2x 3 2x 33x 4 Using Substitution to Factor x y Let 2 7 x y 12 x y z z 7 z 12 2 z 3 z 4 Resubtitute: x y 3 x y 4 Using Grouping to Factor Trinomials 10 x 13x 3 2 10 x 15x 2 x 3 2 10 x 2 15 x 2 x 3 5x 2x 3 2x 3 2x 35x 1