More polynomials help (1 Viewer)

[foram]

Find the cubic equation whos roots are the squares of that of x^3 + 2x + 1 = 0.

How do i solve this?

do i let find the roots in terms of abc, ab + bc + ca, a + b + c? and then do i square it and rearrange it untill i can work out the coefficients of the new cubic? How can i do this?

 

[tommykins]

Let x = a² (a for alpha)

Therefore, a = +-sqrt x (just use sqrt x for now)

P(a) = (sqrtx)^3 + 2sqrtx + 1 = 0

xsqrtx + 2sqrtx = -1

Squaring both sides

(x³ + 4x²+4x) = 1

x³ + 4x² + 4x - 1 = 0 is the new polynomial.

Forgive me if I made any mistakes, I expanded in my head.

 

[foram]

wow, thats such a smart way of doing it! i tried doing it the normal way, which i found in the text book, which was finding the relationship between roots and coefficients, and then it just solved it the long way from there. but your way is about 10 times faster!! you're so smart! XDXD

 

[Slidey]

Find the cubic equation whos roots are the squares of that of x^3 + 2x + 1 = 0.

How do i solve this?

do i let find the roots in terms of abc, ab + bc + ca, a + b + c? and then do i square it and rearrange it untill i can work out the coefficients of the new cubic? How can i do this?

Use substitution:
y=x^2
x=sqrt(y)
Original equation:
x(x^2+2)=-1
x^2(x^4+4x^2+4)=1 (square of original equation in such a way that eliminates odd powers of x)
y(y^2+4y+4)=1 (sub in the value of x)
y^3+4y^2+4y-1=0 (your new equation)

Pretty sure that's a 4unit method, though. I suppose since it's only x^2, you can use it.

In 3unit they likely expect you to use this:
-b/a=l+m+n=0
c/a=lm+ln+mn=2
-d/a=lmn=1

Co-efficients for the new equation:
-b/a=l^2+m^2+n^2
=(l+m+n)^2-2(lm+ln+mn)=0-4=-4

c/a=l^2m^2+l^2n^2+m^2n^2
=(lm+ln+mn)^2-2(l^2mn+lm^2n+lmn^2)=(lm+ln+mn)^2-2lmn(l+m+n)
2^2-2*1*0=4

-d/a=l^2m^2n^2=(lmn)^2=1^2=1

Hence:
x^3+4x^2+4x-1=0

As you can probably tell, this method falls to bits for x^3 or sqrt(x). Thus they teach you the 4unit substitution method.

 

[tommykins]

Really? I found it weird that he's doing transofmration of polynomials in year 11.

 

[cwag]

Let x = a² (a for alpha)

Therefore, a = +-sqrt x (just use sqrt x for now)

P(a) = (sqrtx)^3 + 2sqrtx + 1 = 0

xsqrtx + 2sqrtx = -1

Squaring both sides

(x³ + 4x²+4x) = 1

x³ + 4x² + 4x - 1 = 0 is the new polynomial.

Forgive me if I made any mistakes, I expanded in my head.

haha....four unit really does help in alot of areas ay?

 

[tommykins]

haha....four unit really does help in alot of areas ay?

Tell me about it

 

[foram]

haha....four unit really does help in alot of areas ay?

well screw 2U it's too easy. I'm not comming first in that cause of really careless mistakes. I'll just do 4U math next year, where people in my school will fail, and i will come no.1 and my mark will get scaled to whatever i get in the HSC which will be 98% i hope so it gets scaled to nearly 100. and all i need to do is get 100% in 3U math and i'm set for 4 units of my HSC!

4U math is awesome isn't it, i can't believe i wasted like 30minutes of my life doing those type of polynomial questions the 3U way when there was such an easy and logical 4U method right around the corner!

 

[tacogym27101990]

bit cocky dont u think?
how do u no ull get 98% in the exam??

 

[Trefoil]

No. 2unit can be sort of difficult for 4unit students because it is both pointless (won't count) and boring (meaning it is hard to focus and do well).

 

[mantiswhoprays]

well screw 2U it's too easy. I'm not comming first in that cause of really careless mistakes. I'll just do 4U math next year, where people in my school will fail, and i will come no.1 and my mark will get scaled to whatever i get in the HSC which will be 98% i hope so it gets scaled to nearly 100. and all i need to do is get 100% in 3U math and i'm set for 4 units of my HSC!

4U math is awesome isn't it, i can't believe i wasted like 30minutes of my life doing those type of polynomial questions the 3U way when there was such an easy and logical 4U method right around the corner!

just be careful, i thought that exact thing. don't get me wrong 4U is great fun but you gotta be prepared to work your face off. you gotta cram a lot of stuff in your head in a short amount of time, we do a new idea every lesson. it's intense but it beats 2U.

 

[tommykins]

hahaha let the kid enjoy his cockiness, it'll strive him further until he fails and actually works for it.

 

[foram]

well, right now, i find math to be too slow.

If 4u math is as fast as you say it is, i'll find it much more enjoyable. besides, my schools starts of with harder 3U lol, and thats proberbly the hardest part of 4U math, so once thats over, it'll be a cruise. I also ahve the 4U cambridge book, and there are worked solutions on the site, so i'll be able to get a headstart on it this holiday. =D