We get the term 10x by doing the following: 10x 2 x 5 Q c a Two First term Last term in binomial times in binomial This follows directly from how we found (x 5)2 using the formula. EXAMPLE 1 EXAMPLE 2 The coefficients of the first and last terms in the trinomials are both perfect squares. Formula Factoring a Perfect Square Trinomial a2 2ab b2 (a b)2 a2 2ab b2 (a b)2 Note In order for a trinomial to be a perfect square, two of its terms must be perfect squares. Factor t2 12t 36 completely. Solution We cannot take out a common factor, so let’s see if this follows the pattern of a perfect square trinomial. t 2 12t 36 T T What do you square to get t2? t (t)2 (6)2 What do you square to get 36? 6 Does the middle term equal 2 t 6? Yes. 2 t 6 12t Therefore, t2 12t 36 (t 6)2. Check by multiplying. Factor completely. a) n2 14n 49 b) 4p3 24p2 36p c) 9k2 30k 25 d) 4c2 20c 9 Solution a) We cannot take out a common factor. However, since the middle term is negative and the fi rst and last terms are positive, the sign in the binomial will be a minus () sign. Does this fi t the pattern of a perfect square trinomial? n2 14n 49 T T What do you square to get n2? n (n)2 (7)2 What do you square to get 49? 7 Does the middle term equal 2 n 7? Yes: 2 n 7 14n www.mhhe.com/messersmith SECTION 7.3 Special Factoring Techniques 379
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