S skyd92 New member Joined Jun 29, 2009 Messages 16 Aug 18, 2009 #1 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.
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.
B BigGlenntheHeavy Senior Member Joined Mar 8, 2009 Messages 1,577 Aug 18, 2009 #2 \(\displaystyle Hint: \ (a+b)^{n} \ = \ \sum_{k=0}^{n}\big(\combination_{k}^{n}\big)a^{n-k}b^{k}.\)
S soroban Elite Member Joined Jan 28, 2005 Messages 5,584 Aug 19, 2009 #3 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: .\(\displaystyle \left(x^2 + \tfrac{3}{x}\right)^8\) Click to expand... What is stopping you from expanding it? \(\displaystyle \left(x^2 + \frac{3}{x}\right)^8 \;=\;(x^2)^8 +\; 8(x^2)^7\left(\frac{3}{x}\right) +\; 28(x^2)^6\left(\frac{3}{x}\right)^2 +\; 56(x^2)^5\left(\frac{3}{x}\right)^3 +\; 70(x^2)^4\left(\frac{3}{x}\right)^4 + \hdots\) . . . . . \(\displaystyle = \;x^{16} +\; 8\cdot\! x^{14}\!\cdot\!\frac{3}{x} +\; 28\!\cdot\! x^{12}\!\cdot\!\frac{9}{x^2} +\; 56\!\cdot\! x^{10}\!\cdot\!\frac{27}{x^3} +\; 70\!\cdot\! x^8\cdot\frac{81}{x^4} + \hdots\) . . . . . \(\displaystyle =\;x^{16} + 24x^{13} +\; 252x^{10} + 1512 x^7 +\; \underbrace{5670x^4}_{There!} + \hdots\)
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: .\(\displaystyle \left(x^2 + \tfrac{3}{x}\right)^8\) Click to expand... What is stopping you from expanding it? \(\displaystyle \left(x^2 + \frac{3}{x}\right)^8 \;=\;(x^2)^8 +\; 8(x^2)^7\left(\frac{3}{x}\right) +\; 28(x^2)^6\left(\frac{3}{x}\right)^2 +\; 56(x^2)^5\left(\frac{3}{x}\right)^3 +\; 70(x^2)^4\left(\frac{3}{x}\right)^4 + \hdots\) . . . . . \(\displaystyle = \;x^{16} +\; 8\cdot\! x^{14}\!\cdot\!\frac{3}{x} +\; 28\!\cdot\! x^{12}\!\cdot\!\frac{9}{x^2} +\; 56\!\cdot\! x^{10}\!\cdot\!\frac{27}{x^3} +\; 70\!\cdot\! x^8\cdot\frac{81}{x^4} + \hdots\) . . . . . \(\displaystyle =\;x^{16} + 24x^{13} +\; 252x^{10} + 1512 x^7 +\; \underbrace{5670x^4}_{There!} + \hdots\)
