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

357 Points
15 years ago

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 "(ab)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.