More on Binomial Theorem

[skyd92]

I just don't know how to even proceed with this problem.

Problem: "Find the term involving x^4 in (x^2 + 3/x)^8"

...help.

 

[BigGlenntheHeavy]

\(\displaystyle Hint: \ (a+b)^{n} \ = \ \sum_{k=0}^{n}\big(\combination_{k}^{n}\big)a^{n-k}b^{k}.\)

 

[soroban]

Hello, skyd92!

I just don't know how to even proceed with this problem . . . . Sure you do!

Find the term containing \(\displaystyle x^4\) in: .\(\displaystyle \left(x^2 + \tfrac{3}{x}\right)^8\)

What is stopping you from expanding it?


\(\displaystyle \left(x^2 + \frac{3}{x}\right)^8 \;=\;(x^2)^8 +\; 8(x^2)^7\left(\frac{3}{x}\right) +\; 28(x^2)^6\left(\frac{3}{x}\right)^2 +\; 56(x^2)^5\left(\frac{3}{x}\right)^3 +\; 70(x^2)^4\left(\frac{3}{x}\right)^4 + \hdots\)

. . . . . \(\displaystyle = \;x^{16} +\; 8\cdot\! x^{14}\!\cdot\!\frac{3}{x} +\; 28\!\cdot\! x^{12}\!\cdot\!\frac{9}{x^2} +\; 56\!\cdot\! x^{10}\!\cdot\!\frac{27}{x^3} +\; 70\!\cdot\! x^8\cdot\frac{81}{x^4} + \hdots\)

. . . . . \(\displaystyle =\;x^{16} + 24x^{13} +\; 252x^{10} + 1512 x^7 +\; \underbrace{5670x^4}_{There!} + \hdots\)