The Fundamental Theorem of Algebra

Today's adventure in maths.

The fundamental theorem of algebra certainly sounds very grand. The idea is quite simple, though.

The theorem applies to any polynomial, which is an equation with exponential values like the one in the background of the image above: five x squared, plus six x plus 5 (one day I'll work out how to write an equation properly in here). Often a goal of using a polynomial equation is to find what values of x will make the equation equal zero. The fundamental theorem of algebra says that there will be as many solutions as the number of the highest exponent. So, for the equation mentioned above, there will be two solutions. If the equation were x cubed (x^3) then there would be three solutions.

Aha, you say, but what about an equation like x^2 +1? For that to be zero, x squared would have to equal negative one and no way no how can that happen.

On the contrary, x squared plus 1 does have two solutions, but they're not real solutions. Real numbers are only part of the number system. There are also imaginary numbers. Together, real and imaginary make up the complex number plane. To find the roots of that equation, you have to bust out i, the imaginary number. i has the special property that i squared equals -1!

So, every quadratic equation (an equation with a square as the highest exponent) has two roots, even though you might need to use complex numbers to find them. Every equation with a cube as the highest exponent has three roots, and so on.