Ultra Hard Polynomial Question (1 Viewer)

Dragonmaster262 · Apr 12, 2009

Two of the roots of the equation $x^3+ax^2+b=0$ are reciprocals of each other. Show that $a=b-1/b$

This is the only question in my homework that I don't get so any help will be greatly appreciated.

rocktheyard · Apr 12, 2009

Let the roots of the equation $x^{3} + ax^{2} + 0x + b = 0$ be:

$\alpha$ , $\frac{1}{\alpha }$ , $\beta$ .

Then $\alpha$ $\frac{1}{\alpha }$ (1) + $\alpha$ $\beta$ + $\beta$ $\frac{1}{\alpha }$ = $\frac{c}{a}$ = 0

Also $\alpha$ $\frac{1}{\alpha }$ $\beta$ = $\beta$ = - b

And $\alpha$ + $\frac{1}{\alpha }$ + $\beta$ = - a
Substitute $\beta$ = -b from second equation:
$\therefore$ $\alpha$ + $\frac{1}{\alpha }$ = b - a
So $\frac{\alpha^{2}+1}{\alpha}$ = b - a

Substitute $\beta$ = -b from second equation into first equation now:
1 - b $\alpha$ - $\frac{-b}{\alpha}$ = 0
$\therefore$ 1 - b( $\alpha$ + $\frac{1}{\alpha }$ ) = 0
$\therefore$ 1 - b( $\frac{\alpha^{2}+1}{\alpha}$ ) = 0
Substitute the previous derived equation:
1 - b(b - a) = 0
$\frac{1}{b}$ = b - a
So a = b - $\frac{1}{b}$

$\therefore$ a = b - $\frac{1}{b}$ as required!

LOL Hope you understand that!

Drongoski · Apr 12, 2009

can't believe it! just completed typing out solution about 20 mins ago and I lost it all; maybe pressed wrong button.

Well, here it is again. I know rocktheyard has already posted a solution.

$Let\ roots\ be\ \alpha ,\frac{1\lpha }{\alpha }\ \ and\ \ \beta \\ \\ \therefore \ \ product\ of\ roots:\ \ \alpha \times \frac{1}{\alpha } \times \beta \ =\ -b\ \ \Rightarrow \ \ \beta \ =\ -b\\ \\ \therefore \ \ (-b)^{3}\ +\ a(-b)^{2}\ +\ b\ =\ 0\ \ \ \ (since\ \beta\ is\ a\ root)\ \\ \\ \therefore \ \ -b(b^{2}\ -\ ab\ -\ 1)\ \ =\ \ 0\\ \\ \therefore \ \ \ b^{2}\ -\ ab\ -1\ \ =\ \ 0\\ \\ \therefore \ \ ab\ \ =\ \ b^{2}\ -\ 1\\ \\ \therefore \ \ \ a\ \ \ =\ \ b\ -\ \frac{1}{b}\\ \\ \\ QED$

rocktheyard · Apr 12, 2009

^^^ HAHA

Guess I was too much in a rush to use a simple method like yours.

GUSSSSSSSSSSSSS · Apr 12, 2009

IMO drongoski solution much better...sorry rocktheyard XDD

(and not just cos thats how i did it...)

Dragonmaster262 · Apr 12, 2009

Thanks everyone!

rocktheyard · Apr 12, 2009

Yea lol

I have a problem of always using the first method that comes to mind.
This has been a problem in my maths tests where I end up with many pages of working out, while others do it in like half :mad1:.

GUSSSSSSSSSSSSS · Apr 12, 2009

rocktheyard said:
Yea lol

I have a problem of always using the first method that comes to mind.
This has been a problem in my maths tests where I end up with many pages of working out, while others do it in like half :mad1:.

HAHA yeh, one that always gets me is when ur trying to find the midpoint of the points of intersection in a parametrics/conics q, and you spend half page workin it out and some other kid uses sum of roots and its done in 1 line =[[[ lol

Drongoski · Apr 12, 2009

hey guys . . . you're all great. I'm full of admiration for the rich pool of talent on bos. Like a school of sharks circling and waiting to pounce on every problem that comes your way. IMO most of you are easily E4 material.

gurmies · Apr 15, 2009

Haha GUSS, I know exactly the type of question you are talking about in conics xD. I showed some people in my class that method and now they're all using it! =)

lyounamu · Apr 15, 2009

Drongoski said:
hey guys . . . you're all great. I'm full of admiration for the rich pool of talent on bos. Like a school of sharks circling and waiting to pounce on every problem that comes your way. IMO most of you are easily E4 material.

we love you

annabackwards · Apr 15, 2009

This qusetion was in my 3U half yearly!

Aha woot i did it right ^^

GUSSSSSSSSSSSSS · Apr 15, 2009

annabackwards said:
This qusetion was in my 3U half yearly!

Aha woot i did it right ^^

rank 1 coming your way XDD

GUSSSSSSSSSSSSS · Apr 15, 2009

gurmies said:
Haha GUSS, I know exactly the type of question you are talking about in conics xD. I showed some people in my class that method and now they're all using it! =)

awwww your so kind =]]]

just did a conics q then which uses it....

annabackwards · Apr 15, 2009

GUSSSSSSSSSSSSS said:
rank 1 coming your way XDD

*fingers crossed*

Ultra Hard Polynomial Question (1 Viewer)

Dragonmaster262

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rocktheyard

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rocktheyard

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lyounamu

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