Finding Terms in a Binomial ExpansionDate: 03/27/2001 at 21:07:57 From: Jen Subject: Finding terms for binomial expansion I have a question on my homework that I really do not understand. I am asked to find the fifth term of: (-5m+4n^3)^54 I really have no idea how to solve this problem, and I was wondering if you could show me. I have no problem with doing problems like these up to the eighth power, but once it gets past that, I am totally clueless. Please help. Date: 03/27/2001 at 22:46:13 From: Doctor Roy Subject: Re: Finding terms for binomial expansion Hello Jen, Thanks for writing to Dr. Math. We can use the binomial theorem to solve this problem. The binomial theorem is related to Pascal's triangle, which you can also use to solve this problem. However, I do not think we wish to find the 54th row of the triangle, so let's use the theorem. The theorem states that the r-th term of the expansion of (a + b)^n is given by: C(n,(r-1)) * a^(n-(r-1)) * b^(r-1) where C(n,r) = n! / (r!*(n-r)!) So, for example, the 5th term in the expansion of (x - 2*y)^8 is given by: C(8,4) * a^(8-4) * b^4 = [8! / (4!*4!)] * a^4 * b^4 = 70 * x^4 * (-2*y)^4 = 70 * x^4 * 16*y^4 = 112 * x^4 * y^4 You can solve your problem in a similar fashion: The fifth term of (-5m+4n^3)^54 is given by: C(54,4) * (-5*m)^(54-4) * (4*n^3)^4 The calculation is a bit tedious, but it can be done. I hope this helps. - Doctor Roy, The Math Forum http://mathforum.org/dr.math/ |
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