Factoring out · Martinとコード

Expanding backwards: how to pull a common factor out of terms and put it in front of the brackets.

Expanding backwards

Last time: 3(x + 4) = 3x + 12. Today you will go the other way. You get 3x + 12 and you have to make 3(x + 4) out of it again.

This reverse trip is called factoring out: you find what all the terms have in common and pull it in front of the brackets.

3x + 12: both terms are divisible by three. 3x = 3 · x and 12 = 3 · 4. I factor out the three: 3(x + 4).

What is it for? A factored expression is shorter, easier to calculate with, and in a few lessons you will see that factoring opens equations that otherwise will not go. It is one of the most useful rearrangements in all of algebra.

The best thing about factoring: it has a built-in check. Expand the result back — it must come out exactly as the question. If it does not, you factored wrongly. No guessing, the paper tells you itself.

Ema is packing gift bags for a school event. She has 24 sweets and 18 stickers and every bag has to be exactly the same. “How many bags can I make?” Jonáš calculates: “Something that divides both 24 and 18… six!” Six bags, in each one 4 sweets and 3 stickers. Ema writes it mathematically: 24b + 18s = 6(4b + 3s). “Look, that is factoring,” she laughs. “I pulled the six in front of the brackets and what is in one bag stayed inside.”

💡

After factoring, always expand back. 3(x + 4) = 3x + 12 — does it match the question? Done. It does not? Fix it. This check is free and does not miss.

How to factor out step by step

1. Find the common factor of the numbers. With 6x + 9 ask: what biggest number divides both 6 and 9? Three.

2. Check the letters. When all terms have x, you can factor out x too. With 6x + 9 the second term has no x — only the three is factored out.

3. Divide each term by what you are factoring out, and write the remainders in the brackets. 6x : 3 = 2x, 9 : 3 = 3. Result: 3(2x + 3).

4. Check by expanding. 3 · 2x + 3 · 3 = 6x + 9. It fits.

Factoring out a letter: x² + 5x. Both terms have x: x² = x · x and 5x = x · 5. I factor out x: x(x + 5).

A number and a letter at once: 4x² + 8x. Numbers: four divides them. Letters: both have x. I factor out 4x: 4x(x + 2).

A special situation: when you factor out a whole term, a one is left in the brackets, not a zero. 5x + 5 = 5(x + 1). Check: 5 · x + 5 · 1 = 5x + 5. It fits.

Example 1: factor out from 8x + 12

The biggest number that divides both 8 and 12: four.

I divide the terms: 8x : 4 = 2x and 12 : 4 = 3.

I write: 4(2x + 3).

Check by expanding: 4 · 2x = 8x, 4 · 3 = 12. It fits.

You could factor out two as well: 2(4x + 6). That is not a mistake, but something is left in the brackets that can be factored further. The agreement is: factor out as much as you can.

Example 2: factor out from x² + 7x

The numbers 1 and 7 have no common divisor except one. But the letters: x² = x · x and 7x = x · 7 — both terms have x.

I factor out x: x² : x = x and 7x : x = 7.

Result: x(x + 7).

Check: x · x + x · 7 = x² + 7x. It fits. You can factor out a letter too — and that is exactly what will unlock equations of the type x² = 7x in a few weeks.

Example 3: factor out from 6a² + 9a

Numbers: 6 and 9 are divided by three. Letters: a² and a have a in common.

So I factor out 3a: 6a² : 3a = 2a and 9a : 3a = 3.

Result: 3a(2a + 3).

Check: 3a · 2a = 6a² (three times two, a times a) and 3a · 3 = 9a. It fits.

The method is always double: numbers separately, letters separately, then both together in front of the brackets.

⚠️

When you factor out a whole term, a one is left in the brackets, not a zero: 7x + 7 = 7(x + 1), not 7(x + 0) or 7 · x. A lost one is the most common factoring mistake — a check by expanding always catches it.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. スタート with ten. When it goes well, add more.

Practise powers — x² = x · x is why you can factor x out of x² + 5x.

Practise the questions

On paper

Factor out, expand back, compare. Three times on every line.

Factor out the biggest possible number: 10x + 15, 6a − 8, 12 + 18b. Check each result by expanding.
Factor out a letter: x² + 9x, 5y − y². Check by expanding.
Factor out a number and a letter: 8x² + 12x. Write both search steps (numbers separately, letters separately).
Find the mistake: a classmate wrote 4x + 4 = 4(x + 0). Fix it and explain in one sentence what was lost.

Now you 💪

I look for a common divisor separately in the numbers and separately in the letters.
After factoring out a whole term, a one stays in the brackets.
I check every factoring by expanding back.

✅

Done when: You have factored all the tasks with expanding checks, the one-mistake fixed, and a run of five correct in practice.

What to take from this lesson

Factoring out is expanding backwards: 3x + 12 = 3(x + 4).
Look for the greatest common divisor of the numbers and shared letters.
After factoring out a whole term a one is left: 5x + 5 = 5(x + 1).
The check is built in: expand back and compare.
Factor out as much as you can — nothing shared may be left in the brackets.