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By Harley Flanders; Justin J Price

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Synopsis of elementary results in pure and applied mathematics

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Lectures on the Cohomology of Groups

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11. 13. 15. 17. 19. 21. /29 = 23 efV: = Vx VX2+T = x + l 2a(a2 + 4a + I ) = 2a3 + 8a2 + 1 a3 + b3 = (a + b)3 a-2a-2 = a4 Vx3 + 2x2 + x = x(x + 1 ) (25x)(4x) = 100x x+y = t +L x+z z ( - x)4 = -x4 x4 6. = x2 + 1 xz + x4 8. (x - y)(x + y) = x2 + y2 I 10. YY x +y 12. v'3X+5 + v'2X+T = vsx + 14. W = a-213 16. (x/y2)2 = x/y4 18. x + x + x + x = x4 20. Vx + v'2X = v'3X -2 22. x-2 = x 4. ( r - 6 48 1 . BASIC ALOEBRA 23. 25. 27. 29. 31. (x + l )(y + l)(z + I ) 1 = xyz + x + y + z + 3 4x - x7x2 + I = 4x2 - 7x + I (x + y)3 = x3 + 3xy + y3 x2 + 4x + 8 = I + 2 + 4 = 7 x2 + 2x + 2 24.

Find the coefficient of x3 in the product: 47. (x2 + 3x + 1)(2x - I) 48. x + 6)(x2 + I ) 49. x2(2x - 5)(x + 6) SO. (x + 1 )(2x - 1 )(4x - I ) St. (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.

2 Multiply: (a) (2x - y)(x2 + xy + 3y2) . . (b) (xy + z)(xy2 + y2z2 + y). SOLUTION (a) Multiply each term in the first parentheses by each term in the second, use rules of exponents, then collect similar terms: (2x - y)(x2 + xy + 3y2 ) = 2x x2 - y x2 + 2x xy - y xy + 2x 3y2 - y 3y2 = 2x3 + (-x2y + 2x2y) + ( - xy2 + 6xy2) - 3y 3 2x3 + x2y + 5xy2 - 3y3. (b) (xy + z)(xy2 + y2z2 + y) xy xy2 + z xy2 + xy y2z2 + z y2z2 + xy y + z y = x2y 3 + xy2z + xy3z2 + y2z3 + xy2 + yz. · · • • · · = = • • · · · · Answer (a) 2x3 + x2y + 5xy2 - 3y3 (b) x2y3 + xy2z + xy3z2 + y2z3 + xy2 + yz.

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