Sunday, 13 October 2013

two quadratic polynomials

Let f(x) and g(x) be two quadratic polynomials all of whose coefficients are rational numbers. Suppose f(x) & g(x) have a common irrational roots. Show that g(x) = r.f(x) for some rational number r. 
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Define \(f: \mathbb{R} \rightarrow \mathbb{R}\) by \[f(x)= \begin{cases}(1-\cos x) \sin \left(\frac{1}{x}\right), & x \neq 0 \\ 0, ...