Ask Uncle Colin: Spotting factors

Dear Uncle Colin,

In a recent test, I stumbled across $9x^4+\frac{1}{144x^4}+\frac{1}{2}$, which apparently factorises as ${(3x^2+\frac{1}{12x^2})}^2$. How on earth am I supposed to spot that?!

- Feeling Almost Cheated, That’s Only Reasonable

Hi, FACTOR, and thanks for your message!

I wouldn’t instinctively spot that that factorises – but I would spot that it’s a hot mess.

I’d certainly notice that $9x^4$ is a factor common to the first term and the bottom of the second, and I’d substitute $y=9x^4$ to see if it made things better: now it’s $y+\frac{1}{16y}+\frac{1}{2}$.

The most ugly thing now is the fractions, so I’d try to turn it into one big fraction: $\frac{16y^2+1+8y}{16y}$.

That thing on the top? That’s a quadratic, and your usual Quadratic Factorising Toolkit will tell you it’s $(4y+1{)}^2$.

We end up with $\frac{(4y+1{)}^2}{16y}$. That’s nice, but we made $y$ up, so we should put it back in terms of $x$: $\frac{(36x^4+1{)}^2}{144x^4}$. In fact, the bottom is also a perfect square, so we can make it ${\left(\frac{36x^4+1}{12x^2}\right)}^2$. This is equivalent to your answer, just a little bit neater!

Hope that helps,

Uncle Colin

* Edited 2017-08-09 to fix an apostrophe.