Expansions • Topic 3 of 3

Conditional Identities and Substitutions

What is a Conditional Identity?

A conditional identity is an algebraic relationship that is true only when a specific given condition or constraint is satisfied. The most famous conditional identity used in Grade 9 mathematics is the cubic summation rule under the constraint that the sum of three variables is zero (a + b + c = 0).

Ordinarily, the full identity connecting three cubes is:

a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)

However, if the problem states that a + b + c = 0, look at what happens to the right side of the equation:

  • The entire bracket value on the right becomes 0.
  • Multiplying anything by zero results in zero: 0 × (a² + b² + c² - ab - bc - ca) = 0.
  • The equation simplifies to: a³ + b³ + c³ - 3abc = 0.
  • Moving the negative term to the other side gives: a³ + b³ + c³ = 3abc.

This shortcut is incredibly useful for evaluating large cubic expressions without manually calculating each cube.

Special Products & Applications(a+b+c)²= a²+b²+c²+2ab+2bc+2caSum of three terms squareda²+b²+c²−ab−bc−ca= ½[(a-b)²+(b-c)²+(c-a)²]Always non-negativea³+b³+c³−3abc= (a+b+c)(a²+b²+c²−ab−bc−ca)Useful factorisationNumerical Shortcut using identities:99² = (100−1)² = 10000 − 200 + 1 = 9801101 × 99 = (100+1)(100−1) = 100² − 1² = 9999
Solved Examples
1
Worked Example

Solve a standard problem on Conditional Identities and Substitutions.

Solution

Apply the formula/method shown in the concept section above.

Key Points

  • Understand the definition and properties of Conditional Identities and Substitutions.
  • Study the worked examples and practice similar problems.
  • Always verify your answer using the original conditions.
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.If $a+b+c=0$, then $a^3+b^3+c^3=$
Explanation: $3abc$.
Q2.$a^3+b^3=$
Explanation: Sum of cubes.
Q3.$a^3-b^3=$
Explanation: Difference of cubes.
Q4.If $a+b+c=0$ and $abc=4$, then $a^3+b^3+c^3=$
Explanation: $3\times4=12$.
Practice more MCQs → 📝 Assignment · 35 marks