Show that $\displaystyle f(x) = (x^3 + 1)/(x - 7)$ is order $\displaystyle x^2$

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- Nov 4th 2009, 05:48 PMracewithferrariSeries and sequences
Show that $\displaystyle f(x) = (x^3 + 1)/(x - 7)$ is order $\displaystyle x^2$

- Nov 4th 2009, 06:02 PMDrexel28
- Nov 4th 2009, 06:58 PMracewithferrari
I don't know but my prof said that start by considering (x-7)f(x) and use theorems on the combination of functions.

- Nov 4th 2009, 07:08 PMDrexel28
- Nov 5th 2009, 05:49 AMchisigma
The 'division' between polynomials can be performed as...

$\displaystyle f(x)=\frac{x^{3}+1}{x-7} = \frac{x^{3}-7x^{2}}{x-7} + \frac{7x^{2} +1}{x-7}=$

$\displaystyle = x^{2} + \frac{7x^{2}-49x}{x-7} + \frac{49x+1}{x-7}=$

$\displaystyle = x^{2} +7x + \frac{49x-343}{x-7} + \frac{344}{x-7}= x^{2} + 7x + 49 + \frac{344}{x-7}$

... so that is...

$\displaystyle f(x)= \mathcal{O} \{x^{2}\}$

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$