Algebra

Factorising single brackets

Group common factors and reverse expansion.

Predict, test and explain

Step 1 of 3

Predict

What is 12x² − 6x after taking out the greatest common factor?

6x + 9

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\text{common numerical factor}=3

The tiles are signed algebraic terms. Equal groups expose the common factor. Expanding the bracket checks both coefficients.

Check the working

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Predict

Try it in the simulation. Change one thing at a time.

Explain

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Hints, self-check and connections

Try a self-check

Fixed values, separate from the diagram controls.

Factorise fully into a single bracket.

$15x^2-10x$

Hint

Find the highest common factor of the coefficients and the shared variable factor.

Reveal answer

Take out five times the variable, leaving three times the variable minus two.

$15x^2-10x=5x(3x-2)$

Hint 1

Find the highest common factor of the coefficients and the smallest power of x shared by every term.

Hint 2

Place that common factor outside the bracket. Divide every original term by it to find the terms inside.

Hint 3

Expand your answer to check it. Factorising reverses expansion.

Worked example

This example uses fixed values, separate from the diagram controls.

Factorise 12x² − 18x fully.

The highest common factor of 12 and 18 is 6. Both terms also contain x, so the full common factor is 6x.
Divide each term by 6x: 12x² ÷ 6x = 2x and −18x ÷ 6x = −3.
Write 6x(2x − 3). Expanding gives 12x² − 18x, which checks the answer.

Answer: 6x(2x − 3)

Connect this idea

Watch out: 6(2x² − 3x) is equivalent, but it is not fully factorised because x is still common to both terms inside the bracket.