Cubed Identities in Algebra

🎯 What will you learn in this chapter?

👉 The formulas related to (a + b)³ and (a - b)³, with explanations.

👉 How to solve problems using the formulas for (a + b)³ and (a - b)³.

👉 Other useful identities derived from the formulas for (a + b)³ and (a - b)³.

👉 The derivation and proof of important cubed identities in algebra.

🔹 In algebra, cubed identities are identities in which the highest power involved is 3.

🔹 An identity is an equality that remains true for every permissible value of the variables. Here, the variables in the expressions are a and b.

🔹 These identities are frequently used in algebra and in several other branches of mathematics.

🔹 They are commonly used to simplify algebraic expressions and to factorise expressions.

🔸

∵ (a + b)³

= (a + b)² (a + b)

= (a² + 2ab + b²) (a + b)

= (a² + 2ab + b²).a + (a² + 2ab + b²).b

= a³ + 2a²b + ab² + a²b + 2ab² + b³

= a³ + 3a²b + 3ab² + b³

∴

🔸

∵

or,

∴

🔸

∵

[ ∵ Subtracting 3ab(a+b) from both sides ]

or,

∴

🔸

∵

or,

∴

🔸

∵ (a – b)³

= (a – b)² (a – b)

= (a² – 2ab + b²) (a – b)

= (a² – 2ab + b²).a – (a² – 2ab + b²).b

= a³ – 2a²b + ab² – a²b + 2ab² – b³

= a³ – 3a²b + 3ab² – b³

∴

🔸

∵

or,

∴

🔸

∵

[ ∵ Adding 3ab(a-b) to both sides ]

or,

∴

🔸

∵

or,

∴

✅ Quick Revision

🔹 (a + b)³ = a³ + 3a²b + 3ab² + b³

🔹 (a + b)³ = a³ + b³ + 3ab(a + b)

🔹 a³ + b³ = (a + b)³ - 3ab(a + b)

🔹 a³ + b³ = (a + b)(a² - ab + b²)

🔹 (a - b)³ = a³ - 3a²b + 3ab² - b³

🔹 (a - b)³ = a³ - b³ - 3ab(a - b)

🔹 a³ - b³ = (a - b)³ + 3ab(a - b)

🔹 a³ - b³ = (a - b)(a² + ab + b²)