The Mathematical Pirate’s Guide to Factorising Cubics

“Yarr,” said the Mathematical Pirate. “Ye’ll have plundered a decent calculator, of course?”

“Er… well, I bought it from Argos, but… aye, cap’n! A Casio fx-83 GT PLUS!”

“A fine calculator,” said the Mathematical Pirate. “One that offers you at least three ways to factorise cubics.”

“Really!? I thought you needed the silver one for that.”

“Pirates deal only in gold.”

“Right, right. OK, then, bring it on! I’ve got a cubic here: it’s $2x^3 – 5x^2 – 9x + 18$.”

The FACT button

“An excellent choice. Please type that into your calculator using the red ALPHA button followed by the close bracket to get your $x$s.”

Tappity tap. “Whoa! That’s a big number.”

“Ah. You’ve probably missed the crucial step of telling it what $x$ is. I think $x$ should be 1000.”

“Really? Well, if you say so.”

“Type 1000, then shift, RCL, close bracket and you should see $1000 \rightarrow x$.”

“I do.”

“Now scroll back to the formula and press =.”

“Whoa! That’s also a big number, but at least it’s an integer.”

“A bit short of two billion, if I’m not mistaken. Now press shift then the button with the quotes on it.”

“FACT!” said the student. “It says $2 \times 3 \times 167 \times 997 \times 1997$.”

“A couple of those are fairly close to multiples of 1000, aren’t they?”

“Oh yes! 997 is three less than 1000, and 1997 is three less than 2000.”

“So you’re talking about $x-3$ and $2x-3$?”

“Oh! I see. And if you divide them out, you get 1002, which is $x+2$!”

“Precisely! So it factorises as $(x-3)(2x-3)(x+2)$. Multiply it out if you don’t believe me!.”

Table mode

The Mathematical Pirate carried on. “Alternatively, you can use the table mode.”

“Oh yes! Mode 3! I’ll type the equation in there. Where do we start?”

“Let’s go from -5 to 5, in steps of 0.5 – and we’re looking for zeros in the $f(x)$ column.”

Scroll scroll scroll. “Oh! -2, 1.5 and 3!”

“Which tells you that $(x+2)$, $(x-1.5)$ and $(x-3)$ are all factors – but we don’t like the 1.5, so we can double it.”

“And we get $(x+2)(2x-3)(x-3)$ again!”

Numerical methods

“Yarr!” said the Mathematical Pirate. “You can also use numerical methods to find a root.”

“Oh! Guessing!”

“Well… yes, I suppose so. You can use the trial and improvement you remember from GCSE…”

The student pulled a face.

“… or you can use Newton-Raphson.”

“Never heard of him.”

“Them,” said the Mathematical Pirate. “You need to know the derivative…”

“… $6x^2 – 10x – 9$…”

“And the rule is, you start with a guess, then take away the value of the function there divided by the derivative of the function there.”

“You mean $x_{k+1} = x_{k} – \frac{f(x_k)}{f’(x_k)}$?”

“If you say so,” said the Mathematical Pirate, flustered.

“Let’s start from $x_0 = 0$,” said the student. Tappity tap… “… 2. Then, having used my answer button cleverly, I can repeat it… 1.2, 1.48, 1.4998, then 1.5, which doesn’t move.”

“It’s converged!”

“1.5 is a solution, so $(2x-3)$ is a factor, I can divide that out and get a quadratic.”

“Yarr,” said the Mathematical Pirate.

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.

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