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Hey guys, this is a question which Slide Rule passed on to me. It's not very difficult but I found it interesting for some reason so I thought I'd share it: Show that the remainder when the polynomial P(x) is divided by (x - a)² is P'(a)x + P(a) - a.P'(a)

That kind of attitude is just as bad as the arrogance you are criticizing.

I've done pretty much all past papers back to 1990 so I'm good in that department. I just need to iron out my weaknesses - conics and geometry in particular. My ultimate goal is to get a mark in the 100's, I've been able to get that in several past papers but I just can't be confident about my...

Some of the harder complex number excercises can be cool (e.g. 1993 Q8). There is a small amount of interesting harder polynomial stuff too, oh and also the unusual combinatorics questions.

I geuss the way I'm looking at it is that all of his operations are reversible, so a proof in one direction implies both.

Never mind, you've explained it enough as is. I think it's an agree to disagree situation :p.

There should be some statement you can make to justify a backwards proof. It feels like this is a grammatical (not a mathematical) issue. It's like the way good is favoured over bad, first before last and over before under ---> likewise forwards is favoured over backwards. In conclusion: I hold...

Hmm, thanks for the explanation. I still think one could argue, however, that there's nothing wrong with using such logic when the issues you describe are 'trivial' (as you put it). If all the operations are perfectly reversible the p implies q also necesitates q implies p , does it not (correct...

Given that you're online at the moment I thought I'd ask, what's your crafty method?

I got an interview offer in the post today (UMAT score = 196).

Physics is far superior to chemistry (p.s. I am biased).

i think I might get it. Is it the difference between say: 'a' represents sin²θ+cos²θ = 1 'b' represents statement x² + y² = 1 where if a is true then not(a) is untrue but if b is true then that does not logically imply that...

I think I've missed what the logical slip is with my case 1. For the sake of argument let q be something like sin²θ+cos²θ = 1 which is, at least in the MX1 forum, an irrefutable truth.

For the range of the values of t i think this might be a way to go about it: We know that 2t = 2x ---> t=x Using our locus definition we know that (-1 - √2) ≤ x ≤ (-1 + √2), i.e. the domain of the circle, so I would assume that - (-1 - √2) ≤ t ≤ (-1 +...

Use the definition |z| = √(x² + y²) to give you: x² + y² - 2i(x + iy) + 2t(1+i) = 0 x² + y² - 2ix + 2y + 2t + 2ti = 0 Equating real parts we get: x² + y² + 2y = -2t ......(1) Equating...

No probs 10 chars

Skipping over the n=1 case (and letting S_n represent the proposition), assume true for n= k so: ∫ (0 to 1) x^ke^x dx = a_k + b_ke ...(1) ... then when n= k+1 ∫ (0 to 1) x^k+1e^x dx =...

For that bit you know that 1 &le; e^x &le; e < 3 (for 0 < x < 1) hence: e^x < 3 x^&alpha;e^x &le; 3x^&alpha; (since x &ge; 0) &there4; &int; (0 to 1) x^&alpha;e^x dx < 3&int; (0 to 1) x^&alpha; dx =...

I've never fully understood the issues with the direction of a proof. It seems to be an issue of semantics 'cause both are logically saying the same thing (aren't they?): Case 1: Assume p. p implies q. q is true, &there4; p is true. (as acmillan did) Case 2: q is true. q implies p...

Are you sure you wrote the question out correctly? If you take an example: 1³ + 2³ + 3³ + 4³ + 5³ = 225 I figured there were two ways to read what you wrote: (n⁴)/3 = 5⁴/3 = 208 1/3 or n^4/3 =...