Revision notesGrade 8Mathematics

Cubic Algebraic Identities

Cubic Algebraic Identities explained through a friendly whiteboard conversation, examples, visuals, and practice for school students.

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Concept Overview

(a + b)^3

Key Definitions

(a + b)^3 = a^{3} + 3

Cube of a Difference

Meaning: This identity expands a minus b cubed.

Definition: $(a - b)^3 = a^{3} - 3$ $a^{2}$ $b + 3$ a $b^{2} - b^{3}$ .

$(x - 3)^3 = x^{3} - 9$ $x^{2} + 27$ $x - 27$ .

(a - b)^3 = a^{3} - 3a^{2}b + 3ab^{2} - b^{3}

Sum of Cubes

Meaning: This identity factorises $a^{3} + b^{3}$ .

Definition: $a^{3} + b^{3} = (a + b)$ ( $a^{2} - a$ $b + b^{2}$ ).

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

a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})

Difference of Cubes

Meaning: This identity factorises $a^{3} - b^{3}$ .

Definition: $a^{3} - b^{3} = (a - b)$ ( $a^{2} + a$ $b + b^{2}$ ).

$x^{3} - 27 = x^{3} - 3^3 = (x - 3)$ ( $x^{2} + 3$ $x + 9$ ).

a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})

Common Mistakes

(a + b)^3

Summary

Cubic identities use power 3. Cube of a sum and difference have four terms. Sum of cubes and difference of cubes factorise differently. Signs must be checked carefully. Cubic identities build on square identities. Cubic identities are important tools for expressions with power 3. They help with expansion, factorisation, and future polynomial work. The main challenge is choosing the correct identity and sign pattern.

Related Concepts

Previous concepts Square identities Cubes and cube roots Polynomial multiplication Next concepts Advanced factorisation Polynomials Quadratic equations Related lessons Algebraic expressions Exponents Identities Curious Maths progression Read this article to understand cubic algebraic identities. Use any matching algebra visualizer to see the structure. Practise the same algebra concept in the workspace. Let adaptive practice recommend the next step for this exact concept.

Cubic Algebraic Identities

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Factorisation — Common Factor & Grouping

Factorising Quadratic Trinomials

Concept Overview

You will learn

Key Definitions

Cube of a Sum

Cube of a Difference

Sum of Cubes

Difference of Cubes

Key Formulae

Worked Examples

Visual Explanation

Common Mistakes

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