Dear Uncle Colin,

In a recent test, I stumbled across $9x^4 + \frac{1}{144x^4} + \frac{1}{2}$, which apparently factorises as $\left(3x^2 + \frac{1}{12x^2}\right)^2$. 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.

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

Share

This site uses Akismet to reduce spam. Learn how your comment data is processed.