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10-3: Factoring Trinomials

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.

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10-3: Factoring Trinomials

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  1. 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.

  2. 10 18 10-3: Factoring Trinomials EXAMPLE 1: Factor 10x2 - 27x + 18. 1 x 180 -1 x -180 2 x 90 -2 x -90 1) Take the coefficient off squared-term and multiply it by the constant term. 3 x 60 -3 x -60 4 x 45 -4 x -45 2) List all the factor pairs of 180. You need to include both positive and negative factors. x 180 = 5 x 36 -5 x -36 6 x 30 -6 x -30 -12 + -15 = -27 3) Find the factor-pair that adds to the coefficient of the middle term. This is your “magic pair.” 9 x 20 -9 x -20 10 x 18 -10 x -18 10x2 - 27x + 18 12 x 15 -12 x -15 4) Replace the middle term by putting the variable on both numbers in the magic pair. 10x2 - 12x - 15x + 18 2x(5x - 6) + 3(-5x + 6) 5) Now, you have a 4-nomial. Factor it using the skills from the last section. 2x(5x - 6) - 3(5x - 6) (5x - 6)(2x - 3)

  3. 1 1 -18 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 14t - 36 + 2t2. = 2t2 + 14t - 36 = 2(t2 + 7t - 18) Re-arrange the terms in descending order and pull out GCF. Factor the trinomial. t2 + 7t - 18 1) Take the coefficient off squared-term and multiply it by the constant term (keeping signs of each term.) 1 x -18 -1 x 18 x = -18 2 x -9 -2 x 9 3 x -6 -3 x 6 2) List all the factor pairs of -18. You need to include both positive and negative factors. -2 + 9 = 7 3) Find the factor-pair that adds to the coefficient of the middle term. This is your “magic pair.” t2 + 7t - 18 t2 - 2t + 9t - 18 4) Replace the middle term by putting the variable on both numbers in the magic pair. t(t - 2) + 9(t - 2) 2(t - 2)(t + 9) (t - 2)(t + 9) 5) Now, you have a 4-nomial. Factor it using the skills from the last section. You have to put the GCF back on your answer.

  4. 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 2a2 - 11a + 7. 1) Take the coefficient off squared-term and multiply it by the constant term (keeping signs of each term.) = 2a2 - 11a + 7 1 + 14 = 15 -1 + -14 = -15 1 x 14 -1 x -14 x 2) List all the factor pairs of 14. You need to include both positive and negative factors. 2 7 = 14 2 x 7 -2 x -7 3) Find the factor-pair that adds to the coefficient of the middle term. This is your “magic pair.” -2 + -7 = -9 2 + 7 = 9 Notice there is no magic pair. Therefore the trinomial is a prime trinomial. Answer: prime polynomial

  5. 10-3: Factoring Trinomials HOMEWORK Page 579 #23 - 41 odd

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