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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: unknown id on February 24, 2008, 03:03:19 pm

Title: Spesh prob
Post by: unknown id on February 24, 2008, 03:03:19 pm: could someone please confirm the following question:

Express in partial form:

$1/(x^4+1)$ ?

Thanks
Title: Re: Spesh prob
Post by: Ahmad on February 24, 2008, 03:12:51 pm: Hint: $x^4 + 1 = (x^2 - \sqrt{2}x + 1)(x^2 + \sqrt{2}x + 1)$
Title: Re: Spesh prob
Post by: Collin Li on February 24, 2008, 04:03:30 pm: It's not on the course. You are only expected to break up linear factors. It's such a MHS thing to do (rolls eyes).
Title: Re: Spesh prob
Post by: unknown id on February 24, 2008, 05:31:50 pm: lol i guess it is but its good to be in the know
Title: Re: Spesh prob
Post by: Four on February 24, 2008, 05:32:13 pm: This is from the coordinate geometry worksheet at MHS, right?

Anyway, the only 'trick' here is that you have to realise that $x^4+1 = (x^2+1)^2 - 2x^2$ and use DOPS from there. Then you can split it up into partial fractions, keeping in mind that if the denominator is an unfactorisable second degree polynomial then the numerator has to be in the form Ax + B.

The next part can be solved a number of ways. I sub’d in complex values of x (solving $x^4 + 1 = (x^2 - \sqrt{2}x + 1)(x^2 + \sqrt{2}x + 1) = 0$ for appropriate values of x to sub in) to eliminate variables and then used simultaneous equations to solve for each individual variable; though equating polynomials might have been easier, it would have involved expanding each bracket.
Title: Re: Spesh prob
Post by: unknown id on February 24, 2008, 05:37:24 pm: yeh, do u go to MHS?