Question

find the coefficient of x power -6 in (3x-4/x)power 10
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Answer 1

Answer:

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Answer 2

Answer:

There is a useful notation, the so-called coefficient of operator [xn] to denote the coefficient of xn of a polynomial or a series. With this notation you can do some kind of bookkeeping of the coefficients you need from the binomials.

We obtain

[x13](1+2x)4(2+x)10=[x13](∑j=04(4j)2jxj)(2+x)10=(∑j=04(4j)2j[x13−j])(2+x)10=((43)23[x10]+(44)24[x9])(2+x)10=(32[x10]+16[x9])∑j=010(10j)210−jxj=32(1010)20+16(109)21=352(1)(2)(3)(4)(5)

Comment:

In (1) we expand the left binomial

In (2) we use the linearity of the coefficient of Operator and [xn]xkp(x)=[xn−k]p(x)

In (3) we see that only j=3 and j=4 provide a contribution

In (4) we expand the other binomial expression

In (5) we take the coefficients of x10 and x9

Step-by-step explanation:

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