If , find the value of

Without actually calculating the cubes, find the value of

(–12)³ + (7)³ + (5)³.

Verify:

(i) x³ + y³ = (x + y) (x² + y² – xy)

(ii) x³ – y³ = (x – y) (x² + y² + xy)

If f(x) = ax² + bx + c is a quadratic polynomial such that f(0) = 0 and f(1) = 1. Prove that, f(x) = ax² + (1 – a)x.

If (x² – 1) is a factor of ax⁴ + bx³ + cx² + dx + e. Then, prove that

a – b + c = d – e