Give a similar formula for the polynomial of the n

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Give a similar formula for the polynomial of the ( n - 1)th degree which assumes, when x = a 1 , a 2 , . . . a n , the values α 1 , α 2 , . . . α n . 16. Find a polynomial in x of the second degree which for the values 0, 1, 2 of x takes the values 1 /c , 1 / ( c + 1), 1 / ( c + 2); and show that when x = c + 2 its value is 1 / ( c + 1). ( Math. Trip. 1911.) 17. Show that if x is a rational function of y , and y is a rational function of x , then Axy + Bx + Cy + D = 0. 18. If y is an algebraical function of x , then x is an algebraical function of y . 19. Verify that the equation cos 1 2 πx = 1 - x 2 x + ( x - 1) r 2 - x 3
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[II : 33] FUNCTIONS OF REAL VARIABLES 78 is approximately true for all values of x between 0 and 1. [Take x = 0, 1 6 , 1 3 , 1 2 , 2 3 , 5 6 , 1, and use tables. For which of these values is the formula exact?] 20. What is the form of the graph of the functions z = [ x ] + [ y ] , z = x + y - [ x ] - [ y ]? 21. What is the form of the graph of the functions z = sin x + sin y , z = sin x sin y , z = sin xy , z = sin( x 2 + y 2 )? 22. Geometrical constructions for irrational numbers. In Chapter I we indicated one or two simple geometrical constructions for a length equal to 2, starting from a given unit length. We also showed how to construct the roots of any quadratic equation ax 2 +2 bx + c = 0, it being supposed that we can construct lines whose lengths are equal to any of the ratios of the coefficients a , b , c , as is certainly the case if a , b , c are rational. All these constructions were what may be called Euclidean constructions; they depended on the ruler and compasses only. It is fairly obvious that we can construct by these methods the length mea- sured by any irrational number which is defined by any combination of square roots, however complicated. Thus 4 v u u t s 17 + 3 11 17 - 3 11 - s 17 - 3 11 17 + 3 11 is a case in point. This expression contains a fourth root, but this is of course the square root of a square root. We should begin by constructing 11, e.g. as the mean between 1 and 11: then 17+3 11 and 17 - 3 11, and so on. Or these two mixed surds might be constructed directly as the roots of x 2 - 34 x +190 = 0. Conversely, only irrationals of this kind can be constructed by Euclidean methods. Starting from a unit length we can construct any rational length. And hence we can construct the line Ax + By + C = 0, provided that the ratios of A , B , C are rational, and the circle ( x - α ) 2 + ( y - β ) 2 = ρ 2 (or x 2 + y 2 + 2 gx + 2 fy + c = 0), provided that α , β , ρ are rational, a condition which implies that g , f , c are rational.
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[II : 33] FUNCTIONS OF REAL VARIABLES 79 Now in any Euclidean construction each new point introduced into the figure is determined as the intersection of two lines or circles, or a line and a circle. But if the coefficients are rational, such a pair of equations as Ax + By + C = 0 , x 2 + y 2 + 2 gx + 2 fy + c = 0 give, on solution, values of x and y of the form m + n p , where m , n , p are rational: for if we substitute for x in terms of y in the second equation we obtain a quadratic in y with rational coefficients. Hence the coordinates of all points obtained by means of lines and circles with rational coefficients are expressible by rational numbers and quadratic surds. And so the same is true of the distance
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