medium difficulty5 min

Cube Identities

Cube identities are standard formulas for expanding or factorising expressions involving cubes of binomials and sums or differences of cubes.

In one line: Cube signs follow the first bracket; sum gives minus in the middle factor.

Practice This Concept

Understand it

The key identities are (a+b)³=a³+3a²b+3ab²+b³, (a-b)³=a³-3a²b+3ab²-b³, a³+b³=(a+b)(a²-ab+b²), and a³-b³=(a-b)(a²+ab+b²). Signs must be watched carefully.

Example

x³+8 = x³+2³ = (x+2)(x²-2x+4).

Formula and rule

(a±b)³ formulas; a³+b³=(a+b)(a²-ab+b²); a³-b³=(a-b)(a²+ab+b²).

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Factorise 27m³-8.

1Write it as (3m)³-2³.
2So it becomes (3m-2)(9m²+6m+4).

Method you can reuse

1Recognise whether it is a binomial cube or sum/difference of cubes.
2Identify a and b, write the matching formula, then simplify each term.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

How does a³-b³ factorise?

Think first, then show answer

a³-b³ factorises as (a-b)(a²+ab+b²). The first bracket keeps the minus sign, and the trinomial has all plus signs.

Common wrong answer

Writing a³+b³=(a+b)(a²+ab+b²).

Why it loses marks: The middle term in the second factor must be -ab.

Don't mix this up

Students often put all negative signs in (a-b)³, but the third term +3ab² is positive.

Exam tip

In sum of cubes, the bracket sign is plus and the middle sign in the trinomial is minus; in difference of cubes, these signs reverse.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Exams may ask expansion of a binomial cube or factorisation of expressions like 8x³+125.

1
Enough for 1 mark
Cube identities are standard formulas for expanding or factorising expressions involving cubes of binomials and sums or differences of cubes.
2
Add for 2 marks
Example: x³+8 = x³+2³ = (x+2)(x²-2x+4).
3
Add for 3 marks
The key identities are (a+b)³=a³+3a²b+3ab²+b³, (a-b)³=a³-3a²b+3ab²-b³, a³+b³=(a+b)(a²-ab+b²), and a³-b³=(a-b)(a²+ab+b²).

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