Algebra by Harley Flanders; Justin J Price

By Harley Flanders; Justin J Price

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Examples: 8213 = w = \164 = 4, 253/2 = V(2s)3 = ffe)3 V56 = 19-1 -1- - V9 II -1 9- 112 - V" According to the definition of amln, where m = 1 , we have a lln = Va, _ = . _ 53 = 125, 3· • so a 11n is just the n-th root of a in new clothes. For example, 1 1/5 = _l . 1000 113 10, 36 1 12 6, 2 32 There is a subtle point in the definition of amln. The same rational number might be expressed in two ways. For instance, suppose m/n = p/q. The definition then gives two possibilities, amln yam, aPl q f/(iP.

I + x3)(1 + x4 )(1 + x5)(1 + x6) 54. (I + 2x)(I + 3x2 )(1 + 4x3 )(1 + 5x4) SS. x(2x + 1)(3x + l ) - (x2 + 2)(3x - I ) S6. ( I + x + xz + x3 )3. 9. POLYNOMIALS IN SEVERAL VARIABLES A polynomial in several variables is a sum of terms of the form I, x, y, x2, xy, y2, x3, x2y, xyz, x3y2z, with various coefficients. A term such as axkymzn is called a monomial. Its degree is the sum k + m + n of its exponents. Thus the degree of 5x2y4 is 6; the degree of -8xyz is 3 . The degree of a polynomial is the highest degree of the monomials it contains.

X4 - 6x3 + 2x2 + 5x + 2)(x3 + x + 4) S2. (x3 + 2x2 + 3x + 4)(6x3 + 7x2 - x - 5) S3. ( I + x3)(1 + x4 )(1 + x5)(1 + x6) 54. (I + 2x)(I + 3x2 )(1 + 4x3 )(1 + 5x4) SS. x(2x + 1)(3x + l ) - (x2 + 2)(3x - I ) S6. ( I + x + xz + x3 )3. 9. POLYNOMIALS IN SEVERAL VARIABLES A polynomial in several variables is a sum of terms of the form I, x, y, x2, xy, y2, x3, x2y, xyz, x3y2z, with various coefficients. A term such as axkymzn is called a monomial. Its degree is the sum k + m + n of its exponents. Thus the degree of 5x2y4 is 6; the degree of -8xyz is 3 .

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