gce further mathematics
posted by lee .
The roots of the equation x^4  3x^2 + 5x  2 = 0 are a, b, c, d. By the relation y=x^2, or otherwise, show that a^2, b^2, c^2, d^2 are the roots of the equation y^4  6y^3 + 5y^2  13y + 4 = 0.
Let y = x^2 and substitute that into the x polynomial.
y^2  3y + 5 sqrt y 2 = 0
y^2 3y 2 =  5 sqrt y
Square both sides
y^4  6y^3 + 9 y^2 4y^2 +12y + 4 = 25 y
y^4  6y^3 +5 y^2 13y +4 = 0
Values of a, b, c and d that were roots of the original (x) equation must have their squares also be solutions of the y equation.
If a r is a root of a polynomial p(x) then (xr)p(x)
Thus the first equation is
(1) p(x)=(xa)(xb)(xc)(xd)=x^43x^2+5x2=0
Suppose you now multiply p(x)(x+a)(x+b)(x+c)(x+d). You would get
(2) (x^2a^2)(x^2b^2)(x^2c^2)(x^2d^2)=
(ya^2)(yb^2)(yc^2)(yd^2)=
y^46y^3+5y^213y+4=0
= your second equation.
What this means is that if you were to use the relation y=x^2 in the second one and divide it by the first you shoud get
(x+a)(x+b)(x+c)(x+d)
Try that and see what you get. If it is the case, then the conclusion follows.
I just realized you don't know the roots. When you do the steps in the previous post you should get
(x+a)(x+b)(x+c)(x+d)=x^43x^25x2
If you get this then the conclusion follows. Multiply these terms to see how the coefficients are related to (1) in the previous post.
the 2nd part of the question asks : find the value of s~2 where s~n = a^n + b^n + c^n + d^n. and hence show that s~8 = 6(s~6)  5(s~4) + 62 . how do i do this?
That's just substituting n=2 to get
s2=a^2+b^2+c^2+d^2
In the original equation we have
x^4+px^3+qx^2+rx+s p=0,q=3,r=5,s=2
(p)^22q=a^2+b^2+c^2+d^2=2*3=6
I think the second part is done recursively but I don't see how just at the moment. It might also make use of the result from the first part. I'll check it in just a bit.
What course are you taking where these question arise? It looks like modern algebra to me. Just wonderin'
I'm aware of using a formula of Newton to find the sum of nth powers so I think that's what may be expected here.
I'm interested to know what you're studying so I know what can be taken as known. If you are studying theory of equations, roots of polys, symmetric functions, etc, then there should be something on Newton's recursive formula.
BTW, what is s supposed to be? we have sn=sum of nth powers. What is s here?
I do not know what s stands for.I'm studying gce a level further mathematics
Okay, do you see the thread with Jack above? We're discussing the same problem. I think I'm having some trouble with the notation right now.
I think sn is just the sum of the nth powers of the roots of the poly., but I'm not sure how the result follows even if that's the case.
yes, i'm trying to follow the problem with Jack now
lee, i know you!
you know my bro jackson?!
Respond to this Question
Similar Questions

further mathematics
The roots of the eqn, x^4 + px^3 + qx^2 + rx + s = 0 where p, q, r, s are constants and s does not equal to 0, are a, b, c, d. (i) a^2 + b^2 + c^2 + d^2 = p^2 2q (in terms of p & q) (ii) 1/a + 1/b + 1/c + 1/d = r/s (in terms of r … 
maths
Show that the equation (1) divided by (x+1)  (x)divided by (x2)=0 has no real roots Well, to begin, start with 1/(x+1x). The x's cancel out because they are opposite signs, so now you have 1/1, or just 1. Then, you are dividing … 
Algebra2
Find the polynomials roots to each of the following problems: #1) x^2+3x+1 #2) x^2+4x+3=0 #3) 2x^2+4x5 #3 is not an equation. Dod you omit "= 0" at the end? 
Precalculus
"Show that x^6  7x^3  8 = 0 has a quadratic form. Then find the two real roots and the four imaginary roots of this equation." I used synthetic division to get the real roots 2 and 1, but I can't figure out how to get the imaginary … 
Algebra
Given the roots, 1/2 and 4, find: (A) The quadratic equation 2x^29x+4 ? 
Algebra
The equation x^2+px+q=0, q cannot be equal to 0, has two unequal roots such that the squares of the roots are the same as the two roots. Calculate the product pq. I think the obvious root would be one but the second roots i just can't … 
mathematics
Use the discriminant to determine the number of real roots the equation has. 3x2 – 5x + 1 =0 A. One real root (a double root) B. Two distinct real roots C. Three real roots D. None (two imaginary roots) 
algebra
if a quadratic equation with real coefficents has a discriminant of 10, then what type of roots does it have? 
maths quadratic eqn
one of the roots of the quadratic equation x^2(4+k)x+12=0 fine, 1)the roots of the equation 2)the possible values of k please show your workings thanks in advance. 
mathsequadratic eqn
one of the roots of the quadratic equation x^2(4+k)x+12=0 find, 1)the roots of the equation 2)the possible values of k please show your workings thanks in advance.