Algebra / Cubed IdentitiesCubed ...

Cubed Identities in Algebra

🎯 What will you learn at the end of this chapter?

👉 The formulas related to (a + b)³ and (a − b)³ and their explanations.

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

👉 Other important formulas derived from (a + b)³ and (a − b)³.

👉 The derivation and proof of cubed identities in algebra.

🔹 In algebra, identities whose highest power is 3 are called cubed identities.

🔹 An identity is an equality that remains true for all permissible values of its variables. Here a and b are variables.

🔹 These identities are widely used in algebra and in many other branches of mathematics.

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

🔸 +
∵ (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,
or,
∴

✍ Practice Problems Based on the Identities Above

🔹 01. What is the value of ?

🔹 02. Simplify (2x + 3y)3 + 3(2x + 3y)2(y − 2x) + 3(2x + 3y)(y − 2x)2 + (y − 2x)3 using an identity.

🔹 03. If , find .

🔹 04. If , find .

🔹 05. If x + y = 1, find x3 + y3 + 3xy.

🔹 06. If , find x3 + 27.

🔹 07. If , find .

🔹 08. If a + b = 0, find a3 + b3.

🔹 09. Expand (a + b){(2a − b)2 + (2a − b)(a − 2b) + (a − 2b)2} using an identity.

🔹 10. If , find .

🔹 11. If m + n = − 2, find m3 + n3 − 6mn.

👉 Click here to see the step-by-step solutions to these problems.

🔸 +
∵ (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,
or,
∴

✍ Practice Problems Based on the Identities Above

🔹 12. Find the value of (x + y)3 + (x − y)3.

🔹 13. If x2 − 2x − 1 = 0, find .

🔹 14. Find the simplest value of .

🔹 15. If x3 + 3x2 + 3x = 7, find x3 − 3x2 + 3x.

🔹 16. If , find .

🔹 17. If a = b + c, prove that a3 − b3 − c3 = 3abc.

🔹 18. If a − b = 1 and a3 − b3 = 61, find ab.

🔹 19. If a + b = 5, ab = 4, and a > b, find (a2 + b2)(a3 − b3).

🔹 20. If x = 0.25, find 64x3 + 144x2 + 108x + 30 using an identity.

🔹 21. If a − b = 8, find a3 − b3 − 24ab using an identity.

🔹 22. If , find .

🧠 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)