By Hassi S., Sebestyen Z., Snoo H.

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**Example text**

B/2a. Thus a b2 −b + c = 0. +b 4a2 2a It then follows that c − (b2 /4a) = 0 and that b2 = 4ac. e. x = 1 or x = −3. 3 Trying the two possibilities: f( 13 ) = 0 but f(−3) = −27 + 36 + 9 − 18 = 0. Thus f(x) must factorise as (x + 3)2 (x − b), and comparing the constant terms in the two expressions for f(x) immediately gives b = 2. Hence, x = 2 is the third root. (c) Here f(x) = x4 + 4x3 + 7x2 + 6x + 2. e. f (x) = 4x3 + 12x2 + 14x + 6 = 0. This has to have an integer solution and, by inspection, this is x = −1.

AP . We have also used the fact that, for positive real numbers, if q ≤ r and s ≤ t then qs ≤ rt. But, from part (a), b1 b2 ≤ b1 + b2 2 10 2 . PRELIMINARY ALGEBRA Thus, a1 a2 · · · aP aP +1 aP +2 · · · aP ≤ 1 P2 P = (b1 + b2 )P (2P )2P = b1 + b2 P b1 + b2 2 2P P . This shows that the result is valid for P = 2M+1 if it is valid for P = 2M . But for m = M = 1 the postulated inequality is simply result (a), which was shown directly. Thus the inequality holds for all positive integer values of m.

8 If 2y + sin y + 5 = x4 + 4x3 + 2π, show that dy/dx = 16 when x = 1. e neither can be written explicitly as a function of the other, and so we are forced to use implicit diﬀerentiation. Starting from 2y + sin y + 5 = x4 + 4x3 + 2π implicit diﬀerentiation, and the use of the chain rule when diﬀerentiating sin y with respect to x, gives 2 dy dy + cos y = 4x3 + 12x2 . dx dx When x = 1 the original equation reduces to 2y + sin y = 2π with the obvious (and unique, as can be veriﬁed from a simple sketch) solution y = π.

### A canonical decomposition for linear operators and linear relations by Hassi S., Sebestyen Z., Snoo H.

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