Ask Uncle Colin: Spotting factors | Colin

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} +…

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