Roots of Polynomials

A quadratic has 0, 1 or 2 roots, right? Wrong!

It always has 2, but they might be so-called ‘complex’ numbers which is not a topic in the A level Maths syllabus. The ‘FTA’ is one of the most important and difficult to prove theorems in mathematics and you’ve been using it all the time without realising!

Your A level Maths exam will definitely have questions involving exponentials. You must be confident with exponential growth & decay, solving exponential equations and differentiation & integration of functions involving exponentials and logs.

You are going to solve a lot of quadratic equations in your A level maths exams. You can solve them by rearranging the completed square, using the quadratic formula or factorising. You also need to be able to use algebraic division to factorise cubics.

Gauss was an amazing mathematician. Amongst other things he proved the fundamental theorem of algebra: that a polynomial equation of degree n with complex coefficients always has n roots (the roots can be made up of real numbers and, possibly, pairs of complex conjugate roots). So a quadratic always has 2 roots and a cubic always has 3 roots, etc.

It’s a little known fact that the fundamental theorem of algebra is actually neither fundamental, nor algebraic. All known proofs involve stepping outside the world of algebra into the rather wonderful world of analysis, but that’s another story for another time.    

The A level Maths syllabus doesn’t mention the FTA because it requires an understanding of complex numbers, which are not covered by the maths course.  Actually it’s worse than that, because when learning A level maths (specifically, in the algebra and quadratics section of the pure maths course) you will be told that quadratics have either 0, 1 or 2 roots (which is a lie, they have 0 1 or 2 real roots) and that you can determine which using the discriminant:

b2-4ac=0 1 (repeated real) root

b2-4ac<0 No (real) roots

b2-4ac>0 2 distinct (real) roots

Once you learn about complex numbers, you realise that ‘no real roots’ actually means 2 complex conjugate roots. But wait – quadratics with no roots don’t cross the x axis (we can see this when we sketch their graph), so how do they have roots?  Answering that question involves thinking in 4 dimensions!

KEY POINTS: A level Maths + Algebraic Division + Quadratics + Cubics + Graphs  + Discriminant