Part of ALG-06 — Binomial Theorem

Term Independent of x

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To find the term independent of x (constant term) in (xax^a + 1/xbx^b)^n:

Step 1: Write general term Tr+1T_{r+1} = C(n,r) * xa(nr)x^{a(n-r)} * xbrx^{-br} = C(n,r) * xan(a+b)rx^{an - (a+b)r}. Step 2: Set exponent = 0: an - (a+b)r = 0, so r = ana+b\frac{an}{a+b}. Step 3: If r is a non-negative integer <= n, the independent term is C(n, r).

If r is not an integer, there is no term independent of x.

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