Question

How do you factor `a^4+4b^4` ?

Answer 1

`a^4+4b^4 = (a^2-2ab+2b^2)(a^2+2ab+2b^2)`

Explanation:

This polynomial factors nicely into two quadratic polynomials with integer coefficients:

`a^4+4b^4 = (a^2-2ab+2b^2)(a^2+2ab+2b^2)`

These quadratic factors have no simpler linear factors with Real coefficients. To see that, you can check their discriminants:

`Delta_(a^2-2ab+2b^2) = (-2)^2-4(1)(2) = 4-8 = -4`

`Delta_(a^2+2ab+2b^2) = 2^2-4(1)(2) = 4-8 = -4`

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Footnotes

`a^4+4b^4` is an example of a homogeneous polynomial in two variables, `a` and `b`. That is, all of the terms are of the same degree, namely `4` in this example.

Factoring homogeneous polynomials is very similar to factoring a corresponding polynomial in one variable.

In our example, we could let `t = a/b` and look at how to factor:

`t^4+4 = (t^2-2t+2)(t^2+2t+2)`

Then we could multiply this through by `b^4` to get our original factorisation.

Similarly, when we look at `a^2-2ab+2b^2`, the corresponding polynomial in `a/b` formed by dividing it by `b^2` can be expressed as:

`t^2-2t+2`

When you have a quadratic of the form `At^2+Bt+C` in one variable you are probably familiar with finding the discriminant `Delta = B^2-4AC`. The sign of the discriminant `Delta` allows us to determine whether the quadratic in `t` has Real zeros and therefore linear factors with Real coefficients.

Faced with `Aa^2+Bab+Cb^2` the same discriminant tells us whether this factors into linear factors with Real coefficients.