63 R.6 Factoring Polynomials (b) 2y2 + az - 2z - ay2 = 2y2 - 2z - ay2 + az Rearrange the terms. = 12y2 - 2z2 + 1-ay2 + az2 Group the terms. = 21y2 - z2 - a1y2 - z2 Factor out 2 and -a so that y2 - z is a common factor. = 1y2 - z212 - a2 Factor out y2 - z. (c) 4x3 + 2x2 - 2x - 1 = 14x3 + 2x22 + 1-2x - 12 Group the terms. = 2x212x + 12 - 112x + 12 Factor each group. = 12x + 1212x2 - 12 Factor out 2x + 1. S Now Try Exercises 29 and 31. Be careful with signs here. EXAMPLE 3 Factoring Trinomials Factor each trinomial, if possible. (a) 4y2 - 11y + 6 (b) 6p2 - 7p - 5 (c) 2x2 + 13x - 18 (d) 16y3 + 24y2 - 16y SOLUTION (a) We must find values for integers a, b, c, and d in such a way that 4y2 - 11y + 6 = 1ay + b21cy + d2. FOIL method Using the FOIL method, we see that ac = 4 and bd = 6. The positive factors of 4 are 4 and 1 or 2 and 2. Because the middle term has a negative coefficient, we consider only negative factors of 6. The possibilities are -2 and -3 or -1 and -6. Now we try various arrangements of these factors until we find one that gives the correct coefficient of y. 12y - 1212y - 62 = 4y2 - 14y + 6 Incorrect 12y - 2212y - 32 = 4y2 - 10y + 6 Incorrect 1y - 2214y - 32 = 4y2 - 11y + 6 Correct Therefore, 4y2 - 11y + 6 factors as 1y - 2214y - 32. Factoring Trinomials Factoring is the opposite of multiplication. Multiplication 12x + 1213x - 42 = 6x2 - 5x - 4 Factoring One strategy when factoring trinomials uses the FOIL method in reverse. This strategy requires trial-and-error to find the correct arrangement of coefficients of the binomial factors.
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