We now learn how to find a specific term, or a specific power of \(x\), in a binomial expansion, without writing all of the terms in the expansion.
Find the \(x^4\) term in the expansion of: \[\begin{pmatrix}2x^2 - x\end{pmatrix}^5\]
The idea for answering such questions is to work with the general term of the binomial expansion. For instance, looking at \(\begin{pmatrix}2x^2 - x\end{pmatrix}^5\), we know from the binomial expansions formula that we can write: \[\begin{pmatrix}2x^2 - x\end{pmatrix}^5 = \sum_{r=0}^5\begin{pmatrix}5\\r \end{pmatrix}.\begin{pmatrix}2x^2\end{pmatrix}^{5-r}.(-x)^r\] In this case, the general term would be: \[t_r = \begin{pmatrix}5\\r \end{pmatrix}.\begin{pmatrix}2x^2\end{pmatrix}^{5-r}.(-x)^r\] Locating a specific power of \(x\), such as the \(x^4\), in the binomial expansion therefore consists of determining the value of \(r\) at which \(t_r\) corresponds to that power of \(x\). For \(x^4\) that would mean determining the value of \(r\) at which \(t_r = \begin{pmatrix}5\\r \end{pmatrix}.\begin{pmatrix}2x^2\end{pmatrix}^{5-r}.(-x)^r\) is an \(x^4\) term.
We further explain the method for answering such questions through a series of tutorials. These tutorials deal with more and more complicated expansions. Start working through the exercise, further down, as soon as you feel ready.
To learn the method, let's see how to find the \(x^8\) term in the expansion of: \[\begin{pmatrix}x^2+2\end{pmatrix}^7\] This is explained in the following tutorial.
In this second tutorial we work through another example, in which we find the \(x^7\) term in the expansion of \(\begin{pmatrix}x^2-3x\end{pmatrix}^5\).
In this third tutorial we work through another example, in which we find the \(x^8\) term in the expansion of \(\begin{pmatrix}2x^3+\frac{1}{x}\end{pmatrix}^8\).
In this fourth tutorial we work through another example, in which we find the constant term in the expansion of \(\begin{pmatrix}2x-\frac{3}{x^2}\end{pmatrix}^9\).
Using the method described in the tutorials above, answer each of the following:
We find the following results:
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