Plz solve this problem....Express 1296x12 – 4320x9y2 + 5400x6y4 – 3000x3y6 + 625y8 in the form (a + b)n.

I know that the first term is of the form aⁿ, because, for whatever n is, the first term is _nC₀ (which always equals 1) times aⁿ times b⁰ (which also equals 1). So 1296x¹² = aⁿ. By the same reasoning, the last term is bⁿ, so 625y⁸ = bⁿ. And since there are alternating "plus" and "minus" signs, I know from experience that the sign in the middle has to be a "minus". (If all the signs had been "plusses", then the middle sign would have been a "plus" also. But in this case, I'm really looking for "(a – b)ⁿ".)

I know that, for any power n, the expansion has n + 1 terms. Since this has 5 terms, this tells me that n = 4. So to find a and b, I only have to take the 4th root of the first and last terms of the expanded polynomial:

Then a = 6x³, b = 5y², there is a "minus" sign in the middle, and:

1296x¹² – 4320x⁹y² + 5400x⁶y⁴ – 3000x³y⁶ + 625y⁸ = (6x³ – 5y²)⁴

Don't let the Binomial Theorem scare you. It's just another formula to memorize. A really complicated and annoying formula, I'll grant you, but just a formula, nonetheless. Don't overthink the Theorem; there is nothing deep or meaningful here. Just memorize it, and move on.

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