Higher Level Maths · Algebra · Section 3 of 8

Factors.
Five patterns. Spot, then split.

Factoring isn't really doing anything — it's recognising a pattern. There are five families to know: grouping, difference of two squares (DOTS), sum and difference of cubes, trinomials, and combinations — problems that need two of the above stacked. Spot the pattern, apply the formula, you're done.

14 worked examples · Grouping · DOTS · Cubes · Trinomials · Combinations

Same buttons. Every example.

Three choices on every question. Try it on paper first — that's how it sticks.

✏️ Try it first 📖 Show me ⚡ Just the answer

The working writes itself on as you scroll when you click Show me.

Section One

Grouping

The simplest factoring move: take out what's common. With 2 terms, look for a shared factor across both. With 4 terms, group them in pairs — each pair gets its own common factor pulled out, and then a shared bracket pops out as the final common factor.

Question 1 · Two terms — a single common factor

Factorise ax + ay.

What letter or number appears in every term? Pull it out the front, leaving the rest behind in a bracket.

Question 2 · Four terms — group in pairs, then factor twice

Factorise ax + ay + bx + by.

Four terms is the cue. Group the first two together, the last two together. Factor each pair. The bracket left over should match — that's the common factor for the next round.

Question 3 · Four terms but the order is scrambled — rearrange first

Factorise 2px + 9 − 9x − 2p.

Don't try to group as written — (2px + 9) share nothing. Move the terms around so each pair has a common factor. Tip: put the p terms together and the constant-times-x terms together.

Section Two

Difference of Two Squares (DOTS)

Whenever you see something squared minus something else squared, you can factor it instantly. The pattern is symmetric: the two factors are (first + second) and (first − second).

The pattern — learn it once, use it forever

Difference of two squares.

The whole technique fits on one line:

x² − y² = (x − y)(x + y)

The x and y can be anything — a single letter, a coefficient times a letter, a whole bracketed expression, even a number. As long as you've got square minus square, the formula applies. Watch: only works for minus. x² + y² does not factor over the real numbers.

Question 4 · Basic DOTS

Factorise x² − 25.

What number squared gives 25? Once you can write the second term as a perfect square, the formula does the work.

Question 5 · The "x" can be a whole bracket

Factorise (p + 2q)² − 100.

The first "thing being squared" is (p + 2q). The second is some number whose square is 100. Apply the same formula — just keep the bracket together.

Question 6 · The bracketed part is second — mind the signs in the simplified factors

Factorise 36 − (x − 2y)².

This time the bracket is the one being subtracted. 36 = 6², and (x − 2y)² is already squared. Apply DOTS, then tidy the brackets.

Section Three

Sum and Difference of Two Cubes

Same idea as DOTS, but for cubes — and there are two formulas (one for the sum, one for the difference). Unlike x² + y², the sum of cubes does factor. The key is recognising perfect cubes when you see them.

The two formulas — learn both

Cubes factor too — both sum and difference.

The pattern looks scarier than it is. The first bracket has the same sign as the original; the second has the opposite middle sign and an extra x² and y²:

x³ − y³ = (x − y)(x² + xy + y²)

x³ + y³ = (x + y)(x² − xy + y²)

Mnemonic: "Same, Opposite, Plus" — the first bracket sign matches the original, the middle sign of the second bracket is opposite, and the last sign is always plus.

Memorise the perfect cubes — you'll spot them faster:

Perfect Cubes 1–5

1³1

2³8

3³27

4³64

5³125

Perfect Cubes 6–10

6³216

7³343

8³512

9³729

10³1000

Why does it work? Multiply (x − y)(x² + xy + y²) out: the x³ and −y³ survive, every other term cancels. Try it once and you'll trust the formula forever.

Question 7 · Sum of cubes

Factorise x³ + 8.

What number cubed gives 8? Once you've written it as cube + cube, plug into the sum-of-cubes formula.

Question 8 · Difference of cubes — with a coefficient

Factorise 27p³ − 125.

Both terms are perfect cubes, but the first one's hidden inside a coefficient. 27 = 3³ and 125 = 5³. Rewrite, then apply the formula carefully — remember the "x" in the formula is now a whole expression (3p).

Section Four

Trinomials (Quadratics)

Three-term expressions of the form ax² + bx + c. The reliable method is to split the middle term — find two numbers that multiply to give a × c and add to give b. Then group as before.

Carl's notation — "GN" and "Sum"

Split the middle term.

For ax² + bx + c, the guide number GN = a × c. Find two numbers that multiply to give GN and sum to give b. Use them to split the middle term in two, then group.

Two numbers: product = GN, sum = b.

"Add" or "Sub"? If the two numbers have the same sign, you add their magnitudes (Carl writes "Add"). If they have opposite signs, the magnitudes subtract to give b (Carl writes "Sub"). It's a quick mental check on what kind of pair you're looking for.

Question 9 · Leading coefficient ≠ 1 — split-the-middle, the textbook example

Factorise 3x² − 7x − 6.

a = 3, b = −7, c = −6. Guide number GN = 3 × (−6) = −18. Find two numbers that multiply to −18 and add to −7.

Question 10 · Two-variable trinomial — same recipe, just keep the y

Factorise x² − 7xy + 12y².

The "y²" plays the role of the constant. a = 1, c = 12, so GN = +12. Find two numbers that multiply to +12 and add to −7 — both negative.

Section Five

Combinations — two or more techniques together

Real-world expressions don't always come neatly in one pattern. Often you need to do one technique, then another. The order matters: always pull out a common factor first. Then look at what's left and ask "is this DOTS? a perfect square? a cubic?" — apply whichever fits.

Question 11 · Common factor → DOTS

Factorise 2x² − 50.

Don't try DOTS straight away — 2x² isn't a perfect square. Take out the common factor of 2 first; then what's left will be a clean DOTS.

Question 12 · DOTS on a fourth power — apply DOTS twice

Factorise x⁴ − y⁴.

Learn this rewrite: x⁴ = (x²)². So x⁴ − y⁴ is already square − square. Apply DOTS, then look at the bracket that comes out — it might be DOTS again.

Question 13 · Hidden perfect square + DOTS

Factorise a² + 2ab + b² − 121.

Four terms — tempting to group, but it doesn't work. Look at the first three: a² + 2ab + b². That's a perfect square. Once you spot it, the question is just DOTS.

Question 14 · The hard one — rearrange, then stack two patterns

Factorise x³ − x² − y³ + y².

The natural pairing — (x³ − x²) with (−y³ + y²) — gives x²(x − 1) − y²(y − 1). The two brackets don't match, so grouping dies. Instead: rearrange so all the x's are together, all the y's are together — then use the cube and square formulas, and pull out the shared bracket at the end.

End of Section Three

Five patterns. Stack them when needed.

Grouping for 2 or 4 terms — pair up, factor each pair, take out the shared bracket. DOTS when you see square − square, including when the "square" is a whole expression. Sum and difference of cubes — two formulas, easy to memorise once you've seen them work.

Trinomials — split the middle term using GN = a × c, then group. Combinations — pull out the common factor first, then look at what's left and ask which of the previous four fits. The hardest problems combine two or three patterns. The five tools never change — only the order you use them.

Stuck on a question? Want one looked at before the next test?

Better call Carl.

Drop me the question (paper photo, screenshot, anything) and I'll come back to you with worked-through notes inside 24 hours. That's what these notes — and the grinds — are for.

— Carl

Mathslive.ie