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1. Roots of a polynomial
My problem is with this question:

The two roots which I get are and . Because squaring a number always gives you a positive number, so the square of (as it is a negative number) should be equal to .

My problem is that when I simplify this equation, I get a very complex expression which is difficult to solve further. Does anyone has another easy way to do this question?

Any help would be appreciated!
Last edited by Zishi; 21-06-2012 at 06:30.
2. Re: Roots of a polynomial
What expression do you get?
3. Re: Roots of a polynomial
If the roots of a quadratic equation are and then

Do these help?

If you're not familiar with them then you could call one of the roots and the other then compare coefficients with the expanded form of

4. Re: Roots of a polynomial
I don't see how p can be positive, but since I've had little sleep for the last three nights, my brain's probably on autopilot.
Last edited by ghostwalker; 21-06-2012 at 05:48.
5. Re: Roots of a polynomial
(Original post by ghostwalker)
I don't see how p can be positive, but since I've had little sleep for the last three nights, my brain's probably on autopilot.
The question is may be flawed.

The correct answer is probably p=-6, but why does the question says p>0 and then includes a negative value of p is confusing.
6. Re: Roots of a polynomial
(Original post by Astronomical)
What expression do you get?

(Original post by notnek)
If the roots of a quadratic equation are and then

Do these help?

If you're not familiar with them then you could call one of the roots and the other then compare coefficients with the expanded form of

I didn't know about first rule, using that gives , which in turn gives the value of -6. But the answer key to this question gives the value of p as 3, i.e (c)

Using the second rule - The full form of the equation you've given is . Multiplying this by 3 and comparing the coefficients, this gives and again the value of p as -6.

I think this means that the question is flawed, as pointed out by ghostwalker and raheem.
7. Re: Roots of a polynomial
(Original post by raheem94)
...
Yes, I'm brain dead. I was ignoring the possibility that the roots are complex.

3 does work.
Last edited by ghostwalker; 21-06-2012 at 05:38.
8. Re: Roots of a polynomial
(Original post by Zishi)

I didn't know about first rule, using that gives , which in turn gives the value of -6. But the answer key to this question gives the value of p as 3, i.e (c)

Using the second rule - The full form of the equation you've given is . Multiplying this by 3 and comparing the coefficients, this gives and again the value of p as -6.

I think this means that the question is flawed, as pointed out by ghostwalker and raheem.
http://www.wolframalpha.com/input/?i...B+3+x+%2B3%3D0

As ghostwalker says it does work.

The complex solutions are when p=3.
9. Re: Roots of a polynomial
OK,

Since

then

We reject r=1 as this gives a negative value for p, of -6.

So, the roots of our equation are then

and

Multiplying by 3 as that's the coeff of x^2.
Last edited by ghostwalker; 21-06-2012 at 05:59. Reason: Add a detail.
10. Re: Roots of a polynomial
(Original post by ghostwalker)
Yes, I'm brain dead. I was ignoring the possibility that the roots are complex.

3 does work.
Hmm. Taking gives p as 3. Many thanks. Btw this question has to be done without the use of a calculator - is there any simple way to find values of and ? Or do I have to remember their values?
11. Re: Roots of a polynomial
(Original post by Zishi)
Hmm. Taking gives p as 3. Many thanks. Btw this question has to be done without the use of a calculator - is there any simple way to find values of and ? Or do I have to remember their values?
If you recall the shape of the sin/cos curves, then

PS: It's probably easier to work with the cube roots of unity, if you've covered them.
Last edited by ghostwalker; 21-06-2012 at 05:55.
12. Re: Roots of a polynomial
(Original post by ghostwalker)
If you recall the shape of the sin/cos curves, then

PS: It's probably easier to work with the cube roots of unity, if you've covered them.
Hmm, thanks again. (PRSOM)
13. Re: Roots of a polynomial
Haven't tried solving it myself, but out of curiosity I plugged it into WolframAlpha and apparently p is roughly -10.

http://www.wolframalpha.com/input/?i...%7D%7D%7B6%7D+