98 CHAPTER R Review of Basic Concepts Concepts Examples Perform the operations. 12x2 + 3x + 12 - 1x2 - x + 22 = 12 - 12x2 + 13 + 12x + 11 - 22 = x2 + 4x - 1 1x - 521x2 + 5x + 252 = x3 + 5x2 + 25x - 5x2 - 25x - 125 = x3 - 125 4x2 - 10x + 21 x + 2)4x3 - 2x2 + x - 1 4x3 + 8x2 -10x2 + x -10x2 - 20x 21x - 1 21x + 42 -43 Remainder 4x3 - 2x2 + x - 1 x + 2 = 4x2 - 10x + 21 + -43 x + 2 Multiply. 17 - x217 + x2 = 72 - x2 = 49 - x2 13a + b22 = 13a22 + 213a21b2 + b2 = 9a2 + 6ab + b2 12m- 522 = 12m22 - 212m2152 + 52 = 4m2 - 20m+ 25 R.5 Polynomials Operations To add or subtract polynomials, add or subtract the coefficients of like terms. To multiply polynomials, distribute each term of the first polynomial, multiplying by each term of the second polynomial. To divide polynomials when the divisor has two or more terms, use a process of long division similar to that for dividing whole numbers. Special Products Product of the Sum and Difference of Two Terms 1 x +y2 1x −y2 =x2 −y2 Square of a Binomial 1 x +y22 =x2 +2xy +y2 1x −y22 =x2 −2xy +y2 Factor. p2 + 4pq + 4q2 = 1p + 2q22 9m2 - 12mn + 4n2 = 13m- 2n22 4t2 - 9 = 12t + 3212t - 32 r3 - 8 = 1r - 221r2 + 2r + 42 27x3 + 64 = 13x + 4219x2 - 12x + 162 R.6 Factoring Polynomials Factoring Patterns Perfect Square Trinomial x2 +2xy +y2 = 1x +y22 x2 −2xy +y2 = 1x −y22 Difference of Squares x2 −y2 = 1x +y2 1x −y2 Difference of Cubes x3 −y3 = 1x −y2 1x2 +xy +y22 Sum of Cubes x3 +y3 = 1x +y2 1x2 −xy +y22
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