67 R.6 Factoring Polynomials Factoring by Substitution More complicated polynomials may be factored using substitution. EXAMPLE 7 Factoring by Substitution Factor each polynomial. (a) 1012a - 122 - 1912a - 12 - 15 (b) 12a - 123 + 8 (c) 6z 4 - 13z2 - 5 SOLUTION (a) 1012a - 122 - 1912a - 12 - 15 Replace 2a - 1 with u so that = 10u2 - 19u - 15 12a - 122 becomes u2. = 15u + 3212u - 52 Factor. = 3512a - 1 + 343212a - 12 - 54 Replace u with 2a - 1. = 110a - 5 + 3214a - 2 - 52 Distributive property = 110a - 2214a - 72 Simplify. = 215a - 1214a - 72 Factor out the common factor. (b) 12a - 123 + 8 Replace 2a - 1 with u. = u 3 + 23 Write as a sum of cubes. = 1u + 221u2 - 2u + 42 Factor. = 312a - 12 + 24312a - 122 - 212a - 12 + 44 Replace u with 2a - 1. = 12a + 1214a2 - 4a + 1 - 4a + 2 + 42 Add, and then multiply. = 12a + 1214a2 - 8a + 72 Combine like terms. (c) 6z4 - 13z2 - 5 = 6u2 - 13u - 5 Replace z2 with u. = 12u - 5213u + 12 Factor the trinomial. = 12z2 - 5213z2 + 12 Replace u with z2. S Now Try Exercises 83, 87, and 91. Don’t stop here. Replace u with 2a - 1. Remember to make the final substitution. Factoring Expressions with Negative or Rational Exponents EXAMPLE 8 Factoring Expressions with Negative or Rational Exponents Factor out the least power of the variable or variable expression. Assume all variables represent positive real numbers. (a) 12x-2 - 8x-3 (b) 4m1/2 + 3m3/2 (c) 1y - 22-1/3 + 1y - 222/3 SOLUTION (a) The least exponent of the variable x in 12x-2 - 8x-3 is -3. Because 4 is a common numerical factor, factor out 4x-3.
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