Cubic Identities in Algebra: Solved Problems
🔸 01. What is the value of ?
∵
=
[ ∵ a3 + b3 = (a + b)(a2 − ab + b2) ]
= x2 − xy + y2
🔸 02. Simplify (2x + 3y)3 + 3(2x + 3y)2(y − 2x) + 3(2x + 3y)(y − 2x)2 + (y − 2x)3 using an identity.
∵ (2x + 3y)3 + 3(2x + 3y)2(y − 2x) + 3(2x + 3y)(y − 2x)2 + (y − 2x)3
Let a = (2x + 3y) and b = (y − 2x)
or, a + b = (2x + 3y) + (y − 2x)
or, a + b = 2x + 3y + y − 2x
or, a + b = 4y
∴ (2x + 3y)3 + 3(2x + 3y)2(y − 2x) + 3(2x + 3y)(y − 2x)2 + (y − 2x)3
= a3 + 3a2b+ 3ab2 + b3
= (a + b)3
[ ∵ (x + y)3 = x3 + 3x2y+ 3xy2 + y3 ]
= (4y)3
= 64y3 Ans.
🔸 03. If , find .
∵
or,
[ ∵ Cubing both sides, we get: ]
or,
[ ∵ (x + y)3 = x3 + y3 + 3xy(x + y) ]
or,
or,
or,
🔸 04. If , find .
∵
=
[ ∵ x3 + y3 = (x + y)3 − 3xy(x + y) ]
=
=
= 0
🔸 05. If x + y = 1, find x3 + y3 + 3xy.
∵ x3 + y3 + 3xy
= x3 + y3 + 3xy(x + y)
[ ∵ (x + y) = 1 ]
= (x + y)3
[ ∵ (a + b)3 = a3 + b3 + 3ab(a + b) ]
= 13 [ ∵ (x + y) = 1 ]
= 1 Ans.
🔸 06. If , find x3 + 27.
∵
or,
or, x2 + 9 = 3x
or, (x + 3)(x2 − 3x + 32) = (x + 3) × 0
[ ∵ Multiplying both sides by (x + 3), we get:]
or, x3 + 33 = 0
[ ∵ a3 + b3 = (a + b)(a2 − ab + b2) ]
or, x3 + 27 = 0Ans.
🔸 07. If , find .
∵
or,
[ ∵ Multiplying both sides by 3/2, we get:]
or,
or,
[ ∵ Cubing both sides, we get: ]
or,
[ ∵ (a + b)3 = a3 + b3 + 3ab(a + b) ]
or,
[ ∵ ]
or,
or,
or, Ans.
🔸 08. If a + b = 0, find a3 + b3.
∵ a3 + b3
= (a + b)3 − 3ab(a + b)
[ ∵ x3 + y3 = (x + y)3 − 3xy(x + y) ]
= 03 − 3ab × 0
[ ∵ a + b = 0 ]
= 0
🔸 09. Expand (a + b)((2a − b)2 + (2a − b)(a − 2b) + (a − 2b)2) using an identity.
∵ (a + b){(2a − b)2 + (2a − b)(a − 2b) + (a − 2b)2}
= {(2a − b) − (a − 2b)}{(2a − b)2+ (2a − b)(a − 2b) + (a − 2b)2}
= (2a − b)3 − (a − 2b)3
[ ∵ a3 − b3 = (a − b)(a2 + ab + b2) ]
= 7a3 − 6a2b + 6ab2 − 7b3
🔸 10. If , find .
∵
or,
or,
or,
or,
or,
[ ∵ Cubing both sides, we get: ]
or,
or,
or, Ans.
🔸 11. If m + n = − 2, find m3 + n3 − 6mn.
∵ m + n = − 2
or, (m + n)3 = (− 2)3
or, m3 + n3 + 3nm(m + n) = (− 2)3
[ ∵ (a + b)3 = a3 + b3 + 3ab(a + b) ]
or, m3 + n3 + 3nm(− 2) = − 8
[ ∵ m + n = − 2 ]
or, m3 + n3 − 6nm = − 8
🔸 12. Find the value of (x + y)3 + (x − y)3.
∵ (x + y)3 + (x − y)3
= (x3 + 3x2y + 3xy2 + y3) + (x3 − 3x2y + 3xy2 − y3)
= 2x3 + 6xy2
= 2x(x2 + 3y2) Ans.
🔸 13. If x2 − 2x − 1 = 0, find .
∵ x2 − 2x − 1 = 0
or, x2 − 1 = 2x
[ ∵ Dividing both sides by x, we get:]
or,
or,
or,
or,
[ ∵ Cubing both sides, we get:]
or,
[ ∵ (a − b)3 = a3 − b3 − 3ab(a − b) ]
or,
[ ∵ Substituting, we get:]
or,
or, Ans.
🔹 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 .
