10 3 Factoring Trinomials OBJECTIVE You must factor

10 -3: Factoring Trinomials OBJECTIVE: You must factor quadratic trinomials. There have been many ways created to factor trinomials over the years. The method presented in these slides will work for any trinomial that can be factored. If the method does not work, the trinomial can not be factored. You need to learn these steps. We start with an example. © William James Calhoun, 2001

10 -3: Factoring Trinomials EXAMPLE 1: Factor 10 x 2 - 27 x + 18. 1) Take the coefficient off squaredterm and multiply it by the constant term. 2) List all the factor pairs of 180. You need to include both positive and negative factors. 3) Find the factor-pair that adds to the coefficient of the middle term. This is your “magic pair. ” 4) Replace the middle term by putting the variable on both numbers in the magic pair. 5) Now, you have a 4 -nomial. Factor it using the skills from the last section. 10 x 18 = 180 -12 + -15 = -27 10 x 2 - 27 x + 18 1 x 180 -1 x -180 2 x 90 -2 x -90 3 x 60 -3 x -60 4 x 45 -4 x -45 5 x 36 -5 x -36 6 x 30 -6 x -30 9 x 20 -9 x -20 10 x 18 -10 x -18 12 x 15 -12 x -15 10 x 2 - 12 x - 15 x + 18 2 x(5 x - 6) + 3(-5 x + 6) 2 x(5 x - 6) - 3(5 x - 6)(2 x - 3) © William James Calhoun, 2001

10 -3: Factoring Trinomials This process always works, although you may need to re-arrange terms. You should also pull out any GCFs before beginning trinomial factoring. EXAMPLE 2: Factor 14 t - 36 + 2 t 2. Re-arrange the terms in descending order and = 2 t 2 + 14 t - 36 pull out GCF. 2 + 7 t - 18) = 2(t Factor the trinomial. 1 t 2 + 7 t - 18 1) Take the coefficient off squared-term and multiply it by the constant term (keeping signs of each term. ) 2) List all the factor pairs of -18. You need to include both positive and negative factors. 3) Find the factor-pair that adds to the coefficient of the middle term. This is your “magic pair. ” 4) Replace the middle term by putting the variable on both numbers in the magic pair. 5) Now, you have a 4 -nomial. Factor it using the skills from the last section. 1 x -18 = -18 -2 + 9 = 7 1 x -18 -1 x 18 2 x -9 -2 x 9 3 x -6 -3 x 6 t 2 + 7 t - 18 t 2 - 2 t + 9 t - 18 t(t - 2) + 9(t - 2)(t + 9) 2(t - 2)(t + 9) You have to put the GCF back on your answer. © William James Calhoun, 2001

10 -3: Factoring Trinomials If there is no magic pair for the trinomial, then the polynomial is prime. You can do nothing with prime polynomials for now. EXAMPLE 3: Factor 2 a 2 - 11 a + 7. 1) Take the coefficient off squared-term and 2 - 11 a + 7 = 2 a multiply it by the constant term (keeping signs 1 + 14 = 15 -1 + -14 = -15 of each term. ) 2) List all the factor pairs of 14. You need to include both positive and negative factors. 2 x 7 = 14 3) Find the factor-pair that adds to the coefficient of the middle term. This is your “magic pair. ” 1 x 14 -1 x -14 2 x 7 -2 x -7 2+7=9 -2 + -7 = -9 Notice there is no magic pair. Therefore the trinomial is a prime trinomial. Answer: prime polynomial © William James Calhoun, 2001