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by **Brynlee** » Fri Oct 25, 2013 11:27 am

IB mathematics SL - Binomial Theorem

How can we find the constant term of the expansion of $(x-\frac{3}{x})^4$ ?

Thanks!

by **miranda** » Sat Oct 26, 2013 5:00 am

The general term of a binomial expansion of the form $(a+b)^n$ is given by the following formula:

${n}_C_{r} \ a^{n-r}b^r$

In your question the general term is

${4}_C_{r} \ (x)^{4-r}(-\frac{3}{x})^r=$

${4}_C_{r} \ (x)^{4-r} ((-3)^r)(x^{-r})=$

${4}_C_{r} \ ((-3)^r)x^{4-r}(x^{-r})=$

${4}_C_{r} \ ((-3)^r)x^{4-2r}$ (1)

The constant term must be $x^0$

So, $4-2r=0 \Rightarrow r=2$

Then we can easily calculate the coefficient of this term by plugging in $r=2$ into equation (1)

${4}_C_{2} \ ((-3)^2)x^{4-2(2)}=$

$(6) (9)=54$

Hope these help!!

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