The roots of the eqn x^4 - 3x^2 + 5x - 2 = 0 are a, b, c, d, and a^n + b^n + c^n + d^n is denoted by S~n. The equation of the roots a^2, b^2, c^2, d^2 is y^4 - 6y^3 + 5y^2 - 13y + 4 = 0. State the value of S~2 and hence show that S~8 = 6S~6 - 5S~4 + 62.

I answered this question this morning. Check the page previous to this. Let me know if you have questions about it. The post should be near the bottom of the thread begun by jack.

i do not understand the last part with the S~8 = 6S~6 - 5S~4 + 62.

Ok, that does require a little calculating -and observation.
If you take the poly
x^4+px^3+qx^2+rx+s with roots a,b,c,d and multiply it by x^4-px^3+qx^2-rx+s you get a poly whose terms are x^8,x^6,x^4,x^2,x^0(constant term) and the roots are a^2,b^2,c^2,d^2
The coefficient of the x^6 term is s-2.
If you make the sub. y=x^2 you get a poly y^4+p1y^3+p2y^2+p3y+p4. If you repeat this proces that we did above, i.e. negate the coefficients of the odd power terms and multiply them you get a poly whose terms are y^8,,,y^0 all even numbers. The coefficient of the y^6 term is S-4. If you do the process once more you get S-8.
To evaluate that expression the questioner is asking you to calculate S-4, S-6 and S-8 and verify they satisfy that equation, that's all.
To calculat S-6 I observed that in the first y polynomial the coefficient of y^3 is -s1, of y^2 is s2 and of y is -s3 and that (s1)^3-3s1s2+3s3=S-2 where the lower case s's are the symmetric functions. I'm not sure if you've been asked to calculate that or not. Let me know if this explains matters.

I just noticed a couple typos here. This line;
"To calculat S-6 I observed that in the first y polynomial the coefficient of y^3 is -s1, of y^2 is s2 and of y is -s3 and that (s1)^3-3s1s2+3s3=S-2 "
That should be S-6 not S-2 at the end. The other typos are minor.

i understand better now, thanks

You're welcome, feel free to post if anything's not clear. I should mention that this would be a fairly challenging problem for college students. Nice job doing this here.

To find the value of S~2, we need to evaluate the expression a^2 + b^2 + c^2 + d^2, where a, b, c, and d are the roots of the equation x^4 - 3x^2 + 5x - 2 = 0.

To find the roots of the equation x^4 - 3x^2 + 5x - 2 = 0, you can use a method called factoring or by using the quadratic formula.

Using factoring, we can rewrite the equation as (x^2 - 2)(x^2 - 1) = 0, which gives us two equations:
x^2 - 2 = 0
x^2 - 1 = 0

Solving these equations, we get the roots as x = ±√2 and x = ±1.

Now let's evaluate a^2 + b^2 + c^2 + d^2 using the values of the roots we just found:

a^2 + b^2 + c^2 + d^2 = (√2)^2 + (-√2)^2 + 1^2 + (-1)^2
= 2 + 2 + 1 + 1
= 6

So S~2 = 6.

Now, to show that S~8 = 6S~6 - 5S~4 + 62, we need to find the values of S~4 and S~6.

To find S~4, we need to evaluate the expression a^4 + b^4 + c^4 + d^4. We can use the fact that (a^n + b^n + c^n + d^n) = (a + b + c + d)(a^(n-1) + b^(n-1) + c^(n-1) + d^(n-1)) - (ab + ac + ad + bc + bd + cd)(a^(n-2) + b^(n-2) + c^(n-2) + d^(n-2)) + ... + (-1)^(n)(abcd).

Using this formula, we can find S~4:

S~4 = (a + b + c + d)(a^3 + b^3 + c^3 + d^3) - (ab + ac + ad + bc + bd + cd)(a^2 + b^2 + c^2 + d^2) + 3(abc + abd + acd + bcd) - 4(abcd)

Now, substituting the values of the roots we found earlier, we get:

S~4 = (√2 + (-√2) + 1 + (-1))((√2)^3 + (-√2)^3 + 1^3 + (-1)^3) - ((√2)(-√2) + (√2)(1) + (√2)(-1) + (-√2)(1) + (-√2)(-1) + (1)(-1))((√2)^2 + (-√2)^2 + 1^2 + (-1)^2) + 3(√2*(-√2)*1 + √2*(-√2)*(-1) + √2*1*(-1) + (-√2)*1*(-1)) - 4(√2*(-√2)*1*(-1))

Simplifying this expression, we get:

S~4 = (0)(0) - (0)(6) + 0 - 0
= 0

So S~4 = 0.

Similarly, to find S~6, we need to evaluate the expression a^6 + b^6 + c^6 + d^6:

S~6 = (a + b + c + d)(a^5 + b^5 + c^5 + d^5) - (ab + ac + ad + bc + bd + cd)(a^4 + b^4 + c^4 + d^4) + 5(abc + abd + acd + bcd)(a^2 + b^2 + c^2 + d^2) - 6(abcd)(a + b + c + d)

Substituting the values of the roots, we can find:

S~6 = (0)(0) - (0)(0) + 5(0)(6) - 6(0)
= 0

So S~6 = 0.

Now, substituting the values of S~4 = 0, S~6 = 0, and S~2 = 6 into the equation S~8 = 6S~6 - 5S~4 + 62, we get:

S~8 = 6(0) - 5(0) + 62
= 62

Therefore, we have shown that S~8 = 62.