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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Mathematical Methods CAS => Topic started by: nbalakers24 on February 05, 2010, 05:47:40 pm

Title: division of polynomials
Post by: nbalakers24 on February 05, 2010, 05:47:40 pm: y= 2x^3 - 3x^2 - 29x - 30

find x intercepts using the factor theorem.

i cant get the right solution :S

help appreciated!

:D :D
Title: Re: division of polynomials
Post by: dodgedanpei on February 05, 2010, 05:56:52 pm: If y=0, then x=-2
So then x+2 is a factor.

So divide 2x^3 - 3x^2 - 29x - 30 by x+2 by using long division.
Title: Re: division of polynomials
Post by: nbalakers24 on February 05, 2010, 05:59:59 pm: LOL thanks, that just made me realised my mistake i had (x-1) is a factor
Title: Re: division of polynomials
Post by: superflya on February 05, 2010, 06:38:02 pm: ur linear factor will always be a factor of ur contant which in this case is ur k is 30 so the only possible factors are 1,2,3,5,10,15,30 and their neg values. The linear factor is usually single digits. Once u find the factor, take it's neg and use synthetic division, halfs ur working time.
Title: Re: division of polynomials
Post by: Stroodle on February 05, 2010, 06:43:35 pm: Quote from: superflya on February 05, 2010, 06:38:02 pm
ur linear factor will always be a factor of ur contant which in this case is ur k is 30 so the only possible factors are 1,2,3,5,10,15,30 and their neg values. The linear factor is usually single digits. Once u find the factor, take it's neg and use synthetic division, halfs ur working time.

Factors can also fractions.
Title: Re: division of polynomials
Post by: Ilovemathsmeth on February 05, 2010, 07:04:57 pm: Factors can also be fractions, true. Also remember if one of your terms in the polynomial is missing, you need to add a coefficient of zero.

For example: $P(x) = 2x^3 + 0x^2 + 4x + 5$
Title: Re: division of polynomials
Post by: superflya on February 05, 2010, 07:06:15 pm: usually :P
Title: Re: division of polynomials
Post by: Ilovemathsmeth on February 05, 2010, 07:33:12 pm: Same. I prefer long division.
Title: Re: division of polynomials
Post by: Stroodle on February 05, 2010, 07:45:28 pm: I use both. Synthetic is good just because it's so fast...

Here's a good explanation on how to do it, if you're interested: http://www.purplemath.com/modules/synthdiv.htm
Title: Re: division of polynomials
Post by: superflya on February 05, 2010, 07:45:37 pm: Quote from: andrewloppol on February 05, 2010, 07:29:16 pm
Synthetic division?? I'm only familiar with lonnnng division lol..

http://www.purplemath.com/modules/synthdiv.htm
http://www.purplemath.com/modules/synthdiv2.htm

more efficient and easier to check if u've made any simple calculation errors.
Title: Re: division of polynomials
Post by: the.watchman on February 05, 2010, 07:47:33 pm: Long division is good, but seriously, shortcuts are handy!

Here's an example of what I use (when already known that it is a factor):

Say we're finding $\frac{x^3-3x^2-x+3}{x-3}$ (i know it's easy, but for argument's sake...)

$x^3-3x^2-x+3=(x-3)(x^2+kx-1)$ (the $x^2$ comes from $\frac{x^3}{x}$ and the $-1$ comes from $\frac{3}{-3}$ )

Expand the right-hand side: $x^3-3x^2-x+3=x^3+(k-3)x^2-(3k+1)x+3$

From the above, we can see that $k=0$

So, the quotient is $x^2-1$

I think this is really easy when you practise it a few times, it all falls into place pretty quickly!
Title: Re: division of polynomials
Post by: brightsky on February 05, 2010, 09:05:22 pm: Quote from: the.watchman on February 05, 2010, 07:47:33 pm
Long division is good, but seriously, shortcuts are handy!

Here's an example of what I use (when already known that it is a factor):

Say we're finding $\frac{x^3-3x^2-x+3}{x-3}$ (i know it's easy, but for argument's sake...)

$x^3-3x^2-x+3=(x-3)(x^2+kx-1)$ (the $x^2$ comes from $\frac{x^3}{x}$ and the $-1$ comes from $\frac{3}{-3}$ )

Expand the right-hand side: $x^3-3x^2-x+3=x^3+(k-3)3x^2-(3k+1)x+3$

From the above, we can see that $k=0$

So, the quotient is $x^2-1$

I think this is really easy when you practise it a few times, it all falls into place pretty quickly!

That's cool.
Title: Re: division of polynomials
Post by: the.watchman on February 05, 2010, 09:08:39 pm: Yeah it is, but it may take a while to get your head around it the first few times you try to use it!
Title: Re: division of polynomials
Post by: Stroodle on February 05, 2010, 09:18:48 pm: That's the way our teacher taught us to do it too. She calls it the manipulation method. Saves so much time. I find synthetic division faster when $x^3$ has a large coefficient though.
Title: Re: division of polynomials
Post by: the.watchman on February 05, 2010, 09:47:32 pm: Quote from: Stroodle on February 05, 2010, 09:18:48 pm
That's the way our teacher taught us to do it too. She calls it the manipulation method. Saves so much time. I find synthetic division faster when $x^3$ has a large coefficient though.

Yeah, you're probably right...
Title: Re: division of polynomials
Post by: naved_s9994 on February 05, 2010, 11:22:15 pm: I dont get this exactly.
I thus far have used synthetic for these type problems, and ofcourse am open
to quicker methods.

Can anyone explain this in more depth, or something... thanks
Title: Re: division of polynomials
Post by: kyzoo on February 05, 2010, 11:31:41 pm: http://vcenotes.com/forum/index.php/topic,22087.msg229365.html#msg229365
Title: Re: division of polynomials
Post by: naved_s9994 on February 05, 2010, 11:36:18 pm: Ahh Yea...
I See!

Well, at this stage the synthetic is working great for speed, and
i got the regular method under the belt aswell.
Thanks kyzoo!!

Not a bad technique at all though!