64 CHAPTER R Review of Basic Concepts CHECK 1y - 2214y - 32 = 4y2 - 3y - 8y + 6 FOIL method = 4y2 - 11y + 6 ✓ Original polynomial (b) Again, we try various possibilities to factor 6p2 - 7p - 5. The positive factors of 6 could be 2 and 3 or 1 and 6. As factors of -5 we have only -1 and 5 or -5 and 1. 12p - 5213p + 12 = 6p2 - 13p - 5 Incorrect 13p - 5212p + 12 = 6p2 - 7p - 5 Correct Thus, 6p2 - 7p - 5 factors as 13p - 5212p + 12. (c) If we try to factor 2x2 + 13x - 18, we find that none of the pairs of factors gives the correct coefficient of x. Additional trials are also unsuccessful. 12x + 921x - 22 = 2x2 + 5x - 18 Incorrect 12x - 321x + 62 = 2x2 + 9x - 18 Incorrect 12x - 121x + 182 = 2x2 + 35x - 18 Incorrect This trinomial cannot be factored with integer coefficients and is prime. (d) 16y3 + 24y2 - 16y = 8y12y2 + 3y - 22 Factor out the GCF, 8y. = 8y12y - 121y + 22 Factor the trinomial. S Now Try Exercises 35, 37, 39, and 41. Remember to include the common factor in the final form. Each of the special patterns for multiplication can be used in reverse to obtain a pattern for factoring. Perfect square trinomials can be factored as follows. Factoring Perfect Square Trinomials x2 +2xy +y2 = 1x +y22 x2 −2xy +y2 = 1x −y22 EXAMPLE 4 Factoring Perfect Square Trinomials Factor each trinomial. (a) 16p2 - 40pq + 25q2 (b) 36x2y2 + 84xy + 49 SOLUTION (a) Because 16p2 = 14p22 and 25q2 = 15q22, we use the second pattern shown in the box, with 4p replacing x and 5q replacing y. 16p2 - 40pq + 25q2 = 14p22 - 214p215q2 + 15q22 = 14p - 5q22 NOTE In Example 3, we chose positive factors of the positive first term (instead of two negative factors). This makes the work easier.
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