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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Mathematical Methods CAS => Topic started by: Nato on January 11, 2014, 05:12:49 pm

Title: methods3/4 q's
Post by: Nato on January 11, 2014, 05:12:49 pm: does the remainder theorem work when diving some polynomial by a quadratic?

such as trying to find the remainder when (from essentials textbook) $P(x) = x^{5}-3x^{4}+2x^{3}-2x^{2}+3x+1$ is divided by $x^{2}-1$ . Can this be done without having to do the whole long division?

cheers.
Title: Re: methods3/4 q's
Post by: Orb on January 11, 2014, 05:30:34 pm: Quote from: Nato on January 11, 2014, 05:12:49 pm
does the remainder theorem work when diving some polynomial by a quadratic?

such as trying to find the remainder when (from essentials textbook) $P(x) = x^{5}-3x^{4}+2x^{3}-2x^{2}+3x+1$ is divided by $x^{2}-1$ . Can this be done without having to do the whole long division?

cheers.

There's a thread for 3/4 MM questions btw.

But anyway, I don't believe that the remainder theorem can be used to apply in this circumstance. You can't simply sub in x=1 to find the solution because x^2 is made out of (x+1)(x-1).

So yeah, you should just divide the polynomial through the long division method.
Title: Re: methods3/4 q's
Post by: psyxwar on January 11, 2014, 07:07:48 pm: After looking through my essentials, I found the question:

Given that P(x) can be written in the form $(x^2-1)Q(x)+ax+b$ , where Q(x) is a polynomial and a and b are constants, hence or otherwise, find the remainder when P(x) is divided by $x^2-1$

$P(x)=(x+1)(x-1)Q(x)+ax+b$
$P(1)=a+b=2$
$P(-1)=-a+b=-10$

Adding the two equations we get: $2b=-8$ , from which we get $b=-4$ and $a=6$

Therefore, the remainder, which is ax+b, is 6x-4.
Title: Re: methods3/4 q's
Post by: Nato on January 12, 2014, 02:32:00 pm: this is from essentials.

Well, the graph of $9x^{2}-x^{4}$ is shown. Then we are told to graph (question a) -> $9(x-1)^{2}-(x-1)^{4}$ by 'applying suitable transformations'. I don't think we're supposed to expand it or anything.

I am finding it hard to see the transformations that have taken place. Can anyone provide any hints? Like, could the equation be arranged a certain way?

thanks
Title: Re: methods3/4 q's
Post by: Phy124 on January 12, 2014, 03:23:36 pm: If $f(x) = 9x^2-x^4$ and $g(x) = 9(x-1)^2-(x-1)^4$ then they are related by $g(x) = f(x-1)$ i.e. the graph of $g(x)$ is obtained by translating $f(x)$ one unit to the right.