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Cube Identities

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

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Student-friendly explanation

The key identities are (a+b)^3=a^3+3a^2b+3ab^2+b^3, (a-b)^3=a^3-3a^2b+3ab^2-b^3, a^3+b^3=(a+b)(a^2-ab+b^2), and a^3-b^3=(a-b)(a^2+ab+b^2). Signs must be watched carefully.

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

    x^3+8 = x^3+2^3 = (x+2)(x^2-2x+4).

  4. 4

    Avoid this incomplete answer

    Writing a^3+b^3=(a+b)(a^2+ab+b^2) is wrong because the middle term in the second factor must be -ab.

Definition

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

Example

x^3+8 = x^3+2^3 = (x+2)(x^2-2x+4).

Rule to remember

(a±b)^3 formulas; a^3+b^3=(a+b)(a^2-ab+b^2); a^3-b^3=(a-b)(a^2+ab+b^2).

Memory hook

Cube signs follow the first bracket; sum gives minus in the middle factor.

Examples and method

Worked example

Factorise 27m^3-8. Write it as (3m)^3-2^3. So it becomes (3m-2)(9m^2+6m+4).

Method to apply

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

How CBSE asks it

Exams may ask expansion of a binomial cube or factorisation of expressions like 8x^3+125.

Avoid common mistakes

Common confusion

Students often put all negative signs in (a-b)^3, but the third term +3ab^2 is positive.

Common wrong answer

Writing a^3+b^3=(a+b)(a^2+ab+b^2) is wrong because the middle term in the second factor must be -ab.

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.

Quick check

How does a^3-b^3 factorise?

a^3-b^3 factorises as (a-b)(a^2+ab+b^2). The first bracket keeps the minus sign, and the trinomial has all plus signs.

Answer writing and exam use

1-mark answer

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

2-mark answer

Cube identities are standard formulas for expanding or factorising expressions involving cubes of binomials and sums or differences of cubes. (a±b)^3 formulas; a^3+b^3=(a+b)(a^2-ab+b^2); a^3-b^3=(a-b)(a^2+ab+b^2). x^3+8 = x^3+2^3 = (x+2)(x^2-2x+4).

3-mark answer

The key identities are (a+b)^3=a^3+3a^2b+3ab^2+b^3, (a-b)^3=a^3-3a^2b+3ab^2-b^3, a^3+b^3=(a+b)(a^2-ab+b^2), and a^3-b^3=(a-b)(a^2+ab+b^2). Signs must be watched carefully. (a±b)^3 formulas; a^3+b^3=(a+b)(a^2-ab+b^2); a^3-b^3=(a-b)(a^2+ab+b^2). Factorise 27m^3-8. Write it as (3m)^3-2^3. So it becomes (3m-2)(9m^2+6m+4). Exams may ask expansion of a binomial cube or factorisation of expressions like 8x^3+125. Writing a^3+b^3=(a+b)(a^2+ab+b^2) is wrong because the middle term in the second factor must be -ab.
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