By Abul Hasan Ali Nadwi

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**Sample text**

Hopefully this can be exploited to a contradiction, which is what we are searching for. What do we know about a polynomial that has degree at most n? Well, it has at most n roots. Do we know anything about the roots of p? Well, p is a factor of f , so the roots of p are also roots of f . What are the roots of f ? There are none! ) f is always positive (in fact, it is always at least 1), and hence can have no roots. This means in turn that p can have no roots. What does it mean when a polynomial has no roots?

After all, there are heaps of theorems just waiting to be used: the sum of angles in a triangle add up to 180◦ ; the angle subtended by a chord on an arc is always the same; the angle bisectors are concurrent. We need some angles to start with. With the ‘main’ triangle being ABC, and with all the angle bisectors and circles and stuff revolving about this triangle, it might be best to start with the angles α = ∠BAC, β = ∠ABC, γ = ∠BCA (it is traditional to use Greek letters to denote angles). Of course, we have α + β + γ = 180◦ .

X − an ) + 1 cannot be factorized into two smaller integer polynomials, where the ai s are integers. (Hint: if f (x) factors into two polynomials p(x) and q(x), look at p(x) − q(x). 8. Let f (x) be a polynomial with integer coefﬁcients, and let a, b be integers. Show that f (a) − f (b) can only equal 1 when a, b are consecutive. ) 4 Euclidean geometry Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. H. Hardy, ‘A Mathematicians Apology’ Euclidean geometry was the ﬁrst branch of mathematics to be treated in anything like the modern fashion (with postulates, deﬁnitions, theorems, and so forth); and even now geometry is conducted in a very logical, tightly knit fashion.