🔩Factorising

Anyone can compute $3 \times 4 = 12$; but seeing 12 and knowing it splits into $3 \times 4$ — that is the eye of a factoriser. Algebra has the same two directions: turning $3(x + 2)$ into $3x + 6$ is flowing downstream; packing $3x + 6$ back into brackets as $3(x + 2)$ is today's business, factorising. In Variables & the Function Machine you already played "guess the rule from the output" — factorising is exactly that: working the rule out backwards from the result.

Move one: pull out the common factor

Look at $6x + 9$: the coefficients 6 and 9 share the factor 3, so lift it outside the brackets:

$6x + 9 = 3(2x + 3)$

What's left inside? Each term divided by the common factor: $6x \div 3 = 2x$ and $9 \div 3 = 3$. Check by expanding: $3 \times 2x + 3 \times 3 = 6x + 9$ — a perfect match. The common factor can carry a letter too: $5x^2 + 10x = 5x(x + 2)$.

Move two: split $x^2 + bx + c$

Target: $x^2 + 5x + 6$. To write it as $(x + \square)(x + \square)$, the two numbers in the boxes must have a product of 6 and a sum of 5. Cast around: $2 \times 3 = 6$ and $2 + 3 = 5$ — it's this pair:

$x^2 + 5x + 6 = (x + 2)(x + 3)$

Always verify by expanding: $(x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6$, seam after seam.

Move three: difference of squares

A "square minus square" like $x^2 - 9$ has a fixed split:

$x^2 - 9 = (x + 3)(x - 3)$

Verify by expanding: $(x + 3)(x - 3) = x^2 - 3x + 3x - 9$ — the two middle terms cancel, leaving $x^2 - 9$. Any expression of the form $a^2 - b^2$ splits as $(a + b)(a - b)$. Spotting the squares is step one: $x^2$ is $x \cdot x$ and $9$ is $3^2$ — the homework from Exponents pays off here.

Watch the direction: the difference of squares $x^2 - 9$ splits; $x^2 + 9$ is a sum of squares — for now, remember that it does not split.

Factor Scope

A secret rule hides inside the machine. Feed it numbers and guess the rule!

▶ Input

⚙️? ? ?

Output

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🔍 Reveal rule🎲 New rule

The machine processes the rule "times 3, then plus 2" into the output $3x + 2$ — that is expanding, downstream. Factorising is upstream: staring at $3x + 2$ and asking "which rule built you?" Cycle through several rules in the machine and feel the pair at work — expanding downstream, factorising upstream.

What it's for

Its stage entrance is dramatic: solving equations. Once $(x - 5)(x + 2) = 0$ is factored, the answers walk out on their own — $x = 5$ or $x = -2$. In Quadratic Equations, factorising is the first key to the door.

Check yourself

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Quick quiz

↺ Try again

1. 1. What is 6x + 9 factorised?

   3(2x + 3)3(2x + 9)6(x + 3)

2. 2. What is x² + 7x + 10 factorised?

   (x + 2)(x + 5)(x + 1)(x + 10)(x - 2)(x - 5)

3. 3. How does x² - 16 split?

   (x + 4)(x - 4)(x - 4)²(x + 4)²