61 R.6 Factoring Polynomials Use special products to evaluate each expression. 113. 99 * 101 114. 63 * 57 115. 1022 116. 712 Relating Concepts For individual or collaborative investigation (Exercises 113–116) The special products can be used to perform selected multiplications. On the left, we use 1x + y21x - y2 = x2 - y2. On the right, 1x - y22 = x2 - 2xy + y2. 51 * 49 = 150 + 12150 - 12 = 502 - 12 = 2500 - 1 = 2499 472 = 150 - 322 = 502 - 21502132 + 32 = 2500 - 300 + 9 = 2209 R.6 Factoring Polynomials ■ Factoring Out the Greatest Common Factor ■ Factoring by Grouping ■ Factoring Trinomials ■ Factoring Binomials ■ Factoring by Substitution ■ Factoring Expressions with Negative or Rational Exponents The process of finding polynomials whose product equals a given polynomial is called factoring. Unless otherwise specified, we consider only integer coefficients when factoring polynomials. For example, because 4x + 12 = 41x + 32, both 4 and x + 3 are factors of 4x + 12, and 41x + 32 is a factored form of 4x + 12. A polynomial with variable terms that cannot be written as a product of two polynomials of lesser degree is a prime polynomial. A polynomial is factored completely when it is written as a product of prime polynomials. Factoring Out the Greatest Common Factor To factor a polynomial such as 6x2 y3 + 9xy4 + 18y5, we look for a monomial that is the greatest common factor (GCF) of the three terms. 6x2y3 + 9xy4 + 18y5 = 3y312x22 + 3y313xy2 + 3y316y22 GCF = 3y3 = 3y312x2 + 3xy + 6y22 Distributive property EXAMPLE 1 Factoring Out the Greatest Common Factor Factor out the greatest common factor from each polynomial. (a) 9y5 + y2 (b) 6x2t + 8xt - 12t (c) 141m+ 123 - 281m+ 122 - 71m+ 12 SOLUTION (a) 9y5 + y2 = y219y32 + y2112 GCF = y2 = y219y3 + 12 Original polynomial s CHECK Multiply out the factored form: y219y3 + 12 = 9y5 + y2. ✓ Remember to include the 1.
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