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The key to factoring big and complicated expressions into simpler forms is to remember the factor theorem. The factor theorem says that if we plug a number k into our expression ...

The 'by contradiction' tells us we need to assume the opposite to begin with: 1) Let's assume there is a finite number of prime numbers2) Let P be the largest prime number (the last one) 3) if we multiply...

There can only be two possible outcomes, 'success' and 'fail' (e.g heads or tails) There is a fixed number of trialsThe probabilities of 'success' and 'fail' are constant and stay the same in each trial (...

The best way to approach this question is to solve it using integration by parts.First, recognise that ln(x) = 1 x ln(x), and set du/dx = 1 and v = ln(x). We then find that u = x, and dv/dx = 1/x.With thi...

This is a separable differential equation, so the first step is to separate the two variables. Factorise out the x from the top bracket so that the equation becomes: dy/dx = x(y^2+1)/yThe next step is to ...