23 express 3 2 1 5 and 3 2 1 3 2 1 in the form a b 3

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23. Express ( 3 2 - 1) 5 and ( 3 2 - 1) / ( 3 2 + 1) in the form a + b 3 2 + c 3 4, where a , b , c are rational. [Multiply numerator and denominator of the second expression by 3 4 - 3 2 + 1.] 24. If a + b 3 2 + c 3 4 = 0 , where a , b , c are rational, then a = 0, b = 0, c = 0. [Let y = 3 2. Then y 3 = 2 and cy 2 + by + a = 0 . Hence 2 cy 2 + 2 by + ay 3 = 0 or ay 2 + 2 cy + 2 b = 0 . Multiplying these two quadratic equations by a and c and subtracting, we obtain ( ab - 2 c 2 ) y + a 2 - 2 bc = 0, or y = - ( a 2 - 2 bc ) / ( ab - 2 c 2 ), a rational number, which is impossible. The only alternative is that ab - 2 c 2 = 0, a 2 - 2 bc = 0.
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[I : 19] REAL VARIABLES 40 Hence ab = 2 c 2 , a 4 = 4 b 2 c 2 . If neither a nor b is zero, we can divide the second equation by the first, which gives a 3 = 2 b 3 : and this is impossible, since 3 2 cannot be equal to the rational number a/b . Hence ab = 0, c = 0, and it follows from the original equation that a , b , and c are all zero. As a corollary, if a + b 3 2 + c 3 4 = d + e 3 2 + f 3 4, then a = d , b = e , c = f . It may be proved, more generally, that if a 0 + a 1 p 1 /m + · · · + a m - 1 p ( m - 1) /m = 0 , p not being a perfect m th power, then a 0 = a 1 = · · · = a m - 1 = 0; but the proof is less simple.] 25. If A + 3 B = C + 3 D , then either A = C , B = D , or B and D are both cubes of rational numbers. 26. If 3 A + 3 B + 3 C = 0, then either one of A , B , C is zero, and the other two equal and opposite, or 3 A , 3 B , 3 C are rational multiples of the same surd 3 X . 27. Find rational numbers α , β such that 3 q 7 + 5 2 = α + β 2 . 28. If ( a - b 3 ) b > 0, then 3 s a + 9 b 3 + a 3 b r a - b 3 3 b + 3 s a - 9 b 3 + a 3 b r a - b 3 3 b is rational. [Each of the numbers under a cube root is of the form ( α + β r a - b 3 3 b ) 3 where α and β are rational.] 29. If α = n p , any polynomial in α is the root of an equation of degree n , with rational coefficients. [We can express the polynomial ( x say) in the form x = l 1 + m 1 α + · · · + r 1 α ( n - 1) ,
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[I : 19] REAL VARIABLES 41 where l 1 , m 1 , . . . are rational, as in Ex. 22. Similarly x 2 = l 2 + m 2 a + . . . + r 2 a ( n - 1) , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x n = l n + m n a + . . . + r n a ( n - 1) . Hence L 1 x + L 2 x 2 + · · · + L n x n = Δ , where Δ is the determinant l 1 m 1 . . . r 1 l 2 m 2 . . . r 2 . . . . . . . . . . . . . . . . l n m n . . . r n and L 1 , L 2 , . . . the minors of l 1 , l 2 , . . . .] 30. Apply this process to x = p + q , and deduce the theorem of § 14 . 31. Show that y = a + bp 1 / 3 + cp 2 / 3 satisfies the equation y 3 - 3 ay 2 + 3 y ( a 2 - bcp ) - a 3 - b 3 p - c 3 p 2 + 3 abcp = 0 . 32. Algebraical numbers. We have seen that some irrational numbers (such as 2) are roots of equations of the type a 0 x n + a 1 x n - 1 + · · · + a n = 0 , where a 0 , a 1 , . . . , a n are integers. Such irrational numbers are called algebraical numbers: all other irrational numbers, such as π ( § 15 ), are called transcendental numbers. Show that if x is an algebraical number, then so are kx , where k is any rational number, and x m/n , where m and n are any integers. 33. If x and y are algebraical numbers, then so are x + y , x - y , xy and x/y .
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