Complex Numbers Higher Level Math Q

Leaving Cert Student

z = -3-i
z is a root of the cubic equation az^3 + 22z^2 + bz + 40 = 0

(A)Show that a = 3? - No idea what to do here as plugging a in you still have b as an unknown?
(B)Find b? - given that a=3 I suppose this is grand just plug in z and set equal to zero.
(C)Write down all the roots of the cubic? Do you build up a quadratic using z ans its conjugate, then divide it into your cubic to get the final root?

Thanks, I think the q isnt too bad just the "show that a=3" bit is throwing me off a little?

Sorry I may aswell throw down my differentiation q here too instead of starting another thread and inviting the mods on me... so:

differntiate y=3sin(Q)-sin^3 (Q)
and: y=3cosQ - cos^3 (Q)

Finally: differentiate y=((3+x)/(9-x^2)^.5)
Sorry realize that last one is probably difficult to decipher... anyway ive attempted those and got rather questionable solutions so would greatly appreciate someone else trying them and comparing answers.

Any help or tips as always greatly appreciated!

little sis...

Leaving Cert Student wrote: »

z = -3-i
z is a root of the cubic equation az^3 + 22z^2 + bz + 40 = 0

(A)Show that a = 3? - No idea what to do here as plugging a in you still have b as an unknown?
(B)Find b? - given that a=3 I suppose this is grand just plug in z and set equal to zero.
(C)Write down all the roots of the cubic? Do you build up a quadratic using z ans its conjugate, then divide it into your cubic to get the final root?

Thanks, I think the q isnt too bad just the "show that a=3" bit is throwing me off a little?

Sorry I may aswell throw down my differentiation q here too instead of starting another thread and inviting the mods on me... so:

differntiate y=3sin(Q)-sin^3 (Q)
and: y=3cosQ - cos^3 (Q)

Finally: differentiate y=((3+x)/(9-x^2)^.5)
Sorry realize that last one is probably difficult to decipher... anyway ive attempted those and got rather questionable solutions so would greatly appreciate someone else trying them and comparing answers.

Any help or tips as always greatly appreciated!

Well for the complex numbers question:
(A) all I did was sub in ''-3-i'' for z in the cubic equation and then worked out a by letting the imaginary numbers on one side equal the imaginary on the other side and the real numbers on one side equal the real on the other side. then got a=3. If you don't get me I can type it out.

(B) yehp since you found a you can easily get b....

(C) ehm I think cause you know the root (z=-3-i) then the factor is z+3+i so you divide that into the cubic (the cubic with your answers for a and b subbed in) and then you get the other factors and then you just factorize them and you get their roots I believe.

Scuid Mhór

For your last differentiation question I got 5(6x + 2x^2)^4

Don't know how right it is though.

Fun Scuid Mhór trivia fact: I was trying to work out those complex number questions last night to no avail.

Leaving Cert Student

little sis... wrote: »

Well for the complex numbers question:
(A) all I did was sub in ''-3-i'' for z in the cubic equation and then worked out a by letting the imaginary numbers on one side equal the imaginary on the other side and the real numbers on one side equal the real on the other side. then got a=3. If you don't get me I can type it out.

(B) yehp since you found a you can easily get b....

(C) ehm I think cause you know the root (z=-3-i) then the factor is z+3+i so you divide that into the cubic (the cubic with your answers for a and b subbed in) and then you get the other factors and then you just factorize them and you get their roots I believe.

Ah thanks that makes sense dont know why I didnt cop that... cheers!

Leaving Cert Student

Scuid Mhór wrote: »

For your last differentiation question I got 5(6x + 2x^2)^4

Don't know how right it is though.

Fun Scuid Mhór trivia fact: I was trying to work out those complex number questions last night to no avail.

I think I got that one right as its in the form they asked for in the q...

I got 3/(9-x^2), whole lot of substitution, logarithms and internal double derivatives going on in that one!

BeanbagBallbag

=

Your answer isn't correct (if I have your y=... right).

You can get the above answer by simply bringing the 9-x^2 up and having it to the power of -1/2. Then it is a simple product rule question. You should try it yourself and see if you can get the end result.

Leaving Cert Student

BeanbagBallbag wrote: »

=

Your answer isn't correct (if I have your y=... right).

You can get the above answer by simply bringing the 9-x^2 up and having it to the power of -1/2. Then it is a simple product rule question. You should try it yourself and see if you can get the end result.

Oh I am really sorry, the whole thing is to the log base e. My bad I left it out.

Managed to find the solution though for anyone who wants a look!

http://educate.ie/solutions/HL%20Solutions/HL%20SP4%20Solutions/HLSP4Soln(P1Q7).pdf