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Factor Perfect Square Trinomials When factoring a trinomial, we can always use the techniques that we have previously discussed. However, recognizing a perfect square trinomial can be quite useful. A perfect square trinomial’s first and last terms are perfect squares (4, 9, 16, 25, 36,....) and the middle term is twice the square roots of the first and last terms. a2 + 2ab + b2 = (a + b)2 a2 – 2ab + b2 = (a – b)2 It sounds and looks difficult but is actually a shortcut. We just need to use the square roots of the first and last terms, check the middle term through multiplication, and use the middle term’s sign. Example 1: Factor: x2 – 6x + 9 The first and last terms are perfect squares: x2 = x·x and 9 = 3·3 Check the middle term: 2·x·3 = 6x Use the middle term’s sign: negative (x – 3)2 To check the answer, multiply: (x – 3)2 = (x – 3)(x – 3) x(x – 3) – 3(x – 3) x2 – 3x – 3x + 9 x2 – 6x + 9 Correct Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0) Example 2: Factor: 49a2 + 70ab + 25b2 The first and last terms are perfect squares: 49a2 = 7a·7a and 25b2 = 5b·5b Check the middle term: 2·7a·5b = 70ab Use the middle term’s sign: positive (7a + 5b)2 To check the answer, multiply: (7a + 5b)2 = (7a + 5b)(7a + 5b) 7a(7a + 5b) + 5b(7a + 5b) 49a2 + 35ab + 35ab + 25b2 49a2 + 70ab + 25b2 Correct Example 3: Factor: 4y2 – 60y + 81 The first and last terms are perfect squares: 4y2 = 2y·2y and 81 = 9·9 Check the middle term: 2·2y·9 = 36y ≠ 60y This is not a perfect square trinomial Try another method: Multiply first and last numbers: 4 · 81 = 324 Factors of 324: 1,324; 2,162; 3,108; 4,81; 6,54; 9,36; 12,27; 18,18 Add to get -60. Since the product is positive, both numbers must be negative: -1+(-324) = -325 No -2+(-162) = -164 No -3+(-108) = -111 No -4+(-81) = -85 No -6+(-54) = -60 Yes 4y2 – 6y – 54y + 81 Rewrite and expand -60y 2 4y – 6y – 54y + 81 Split the polynomial in half 2y(2y – 3) – 27(2y – 3) Factor the GCF from each side (2y – 3)(2y – 27) Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0) Example 3: Factor: z2 + 12z – 36 The last term is negative, so it is not a perfect square This is not a perfect square trinomial Try another method: Factors of 36: 1,36; 2,18; 3,12; 4,9; 6,6 Add to get 12. Since the product is -36, one number must be negative: -1+36 = 35 or -36+1 = -35 No -2+18 = 16 or -18+2 = -16 No -3+12 = 9 or -12+3 = -9 No -4+9 = 5 or -9+4 = -5 No Since no set of factors add to 12, the trinomial cannot be factored. Prime Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)