53 R.5 Polynomials (d) 41x2 - 3x + 72 - 512x2 - 8x - 42 = 4x2 - 413x2 + 4172 - 512x22 - 51-8x2 - 51-42 = 4x2 - 12x + 28 - 10x2 + 40x + 20 Multiply. = -6x2 + 28x + 48 Add like terms. S Now Try Exercises 23 and 25. As shown in Examples 2(a), (b), and (d), polynomials in one variable are often written with their terms in descending order (or descending degree). The term of greatest degree is first, the one of next greatest degree is next, and so on. Multiplication One way to find the product of two polynomials, such as 3x - 4 and 2x2 - 3x + 5, is to distribute each term of 3x - 4, multiplying by each term of 2x2 - 3x + 5. 13x - 4212x2 - 3x + 52 = 3x12x2 - 3x + 52 - 412x2 - 3x + 52 Distributive property = 3x12x22 + 3x1-3x2 + 3x152 - 412x22 - 41-3x2 - 4152 Distributive property again = 6x3 - 9x2 + 15x - 8x2 + 12x - 20 Multiply. = 6x3 - 17x2 + 27x - 20 Combine like terms. Another method is to write such a product vertically, similar to the method used in arithmetic for multiplying whole numbers. 2x2 - 3x + 5 3x - 4 -8x2 + 12x - 20 -412x2 - 3x + 52 6x3 - 9x2 + 15x 3x12x2 - 3x + 52 6x3 - 17x2 + 27x - 20 Add in columns. Place like terms in the same column. EXAMPLE 3 Multiplying Polynomials Multiply 13p2 - 4p + 121p3 + 2p - 82. SOLUTION Write like terms in columns. This process is an ordered method of applying the distributive property. 3p2 - 4p + 1 p3 + 2p - 8 -24p2 + 32p - 8 -813p2 - 4p + 12 6p3 - 8p2 + 2p 2p13p2 - 4p + 12 3p5 - 4p4 + p3 p313p2 - 4p + 12 3p5 - 4p4 + 7p3 - 32p2 + 34p - 8 Add in columns. S Now Try Exercise 37. The FOIL method is a convenient way to find the product of two binomials. The memory aid FOIL (for First, Outer, Inner, Last) gives the pairs of terms to be multiplied when distributing each term of the first binomial, multiplying by each term of the second binomial. Distributive property
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