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By Hiram Paley

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Use the trigonometric identity sin2 (θ) + cos2 (θ) = 1 to show that sin(θ) is a root of the polynomial x4 + (α2 − 2β − 2)x2 + (β2 + 2β + 1 − α2 ). 14. Here is a table of some of the values of the function f(x). x: −23 −18 −13 f(x): −51 −36 −21 −8 −6 −3 2 9 24 7 39 12 17 54 69 Explain why this table can be produced by a linear function, and then find the linear function that can produce this table. 15. Here is a table of some of the values of the function g(x). x: −5 g(x): 89 −3 31 −1 1 −3 −13 3 5 1 39 7 101 9 187 11 297 (a) Explain why this table cannot be produced by a linear function but can be produced by a quadratic function.

Every finite group with an odd number of elements is solvable. 6. Every finite group arises as the Galois group of a finite field extension of the rational numbers. Some of the preceding statements are obviously true, and some are obviously false. For some of the others, we may not yet understand what they are talking about. In fact, for statements 4 and 6, mathematicians have been unable to determine whether or not these statements are true. However, our inability to determine whether or not a mathematical statement is true does not change the fact that it must be true or false but cannot be both.

Find real numbers a, b, and c such that 11x2 − 2x + 4 a bx + c = + 2 . x3 + x x x +1 20. When decomposing a rational function into a sum of partial fractions, all the denominators we use are powers of polynomials that cannot be factored any further. A similar type of decomposition can be done for rational numbers, as we can decompose them into sums of fractions such that all the denominators are powers of prime numbers. To illustrate this, find integers A and B such that 43 A B = + . 20 4 5 As was the case with partial fractions, one should begin by multiplying both sides of the equation by the denominator of the left-hand side of the equation.

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