polynomial question please help (1 Viewer)

[FD3S-R]

The polynomial P(x)=x^3+ax^2+bx+6 (a,b real) gas (1-i) as one zero. Find a and b and hence factorise P(x) over the complex field C.

i know u have to use the complex root conjugate, i can find a and b, but i just cant seem to factorise.

thanks

 

[Will Hunting]

Make sure the question read find "real" a and b, otherwise complex conj. doesn't apply

If 1 +- i are roots, then [x-(1-i)], [x-(1+i)] are factors
i.e. x^2 - 2x + 2 a factor

Now, divide your p'nomial with known a,b by x^2 - 2x + 2

P(x) = (x^2 + 2x + 2)(x + 3) should be it

P(x) = [(x+1) - i][(x+1) + i](x + 3) Over C

You got a = 1, b = -4, right?

 

[ngai]

Make sure the question read find "real" a and b, otherwise complex conj. doesn't apply

his question does say "(a,b real)"
or did he edit it after u replied?
anyway, you get 2 roots 1-i and 1+i
so product of roots is (1-i)(1+i)r = -6, and so r = -6/2 = -3 (where r is the 3rd root of our degree 3 poly)
then, like will said, P(x) = [(x+1) - i][(x+1) + i](x + 3) Over C
expand first 2 factors to get the factorisation over real
expand the lot to get a,b

 

[Will Hunting]

Edited after

Ya, factorising to find a,b like ngai did, rather than the reverse, is probably the best way to go. You must've solved P(1-i) = 0, or P(1+i) = 0, for a,b, equating real and imaginary parts, to have found a,b before factorising.

Also, to build on ngai's use of root-coeff. relations, a can, alternatively, be found by noting a = -(sum of roots), then b found by solving P(-3) = 0. This saves having to expand your factorised p'nomial for a,b.

 

[Slidey]

Edited after

The edited message pops up if you edit your post more than 2 minutes after posting it. There is no such edited message, and you posted a while after the the thread starter...

There is no way he could have edited it after you replied.

 

[Will Hunting]

The edited message pops up if you edit your post more than 2 minutes after posting it. There is no such edited message, and you posted a while after the the thread starter...

There is no way he could have edited it after you replied.

Ever heard of a thing called a "life"? Why don't you get one, rather than canvassing pointless trivialities and colouring a Maths forum with personal attacks better suited to a primary school playground? If you want to carry on like a 6-year old, however, I can solemnly asseverate that I issued a response to his message "immediately" after it was posted and that, by your argument (which I have no reason to trust, by the way), he, in fact, edited before 2 mins had elapsed. I really haven't the time for infantile jabs like that, though. I guess I was too wrapped up in the "purpose" of this thread, that being to offer to the discussion at hand, something you obviously don't have the brains, nor the maturity to have done. (Irony/hypocrisy not lost; it's OK in response if you sent 1st)