Polynomial Questions (1 Viewer)

[Amaze]

The cubic equation x^3 + kx + 1 = 0, where k is a constant, has roots α, β and γ. For each positive integer n, S_n = αⁿ+βⁿ+γⁿ.
i/ State the value of S₁ and express S₂ in terms of k.
ii/ Show that for all n, S_n+3 + kS_n+1 + S_n = 0
iii/ Hence, or otherwise, express α⁴+β⁴+γ⁴ in terms of k.

I'm unsure on how to approach part ii/. Any help is appreciated! Thanks

Also, for questions where they state something like: P(x) is a polynomial equation which when divided by (ax+b) leaves a remainder of ___, and when divided by (cx+d) leaves a remainder of ___. Find the remainder when P(x) is divided by a certain quadratic. How would I do these kinds of questions?

 

[TimeGoesTooFast]

I might be wrong but this is what I think (replaced y with c):

ii) LHS= a^n+3 + b^n+3 + c^n+3 + k(a^n+1 + b^n+1 + c^n+1) + a^n + b^n + c^n (expand and simplify. Note: a^n+3 = a^3 x a^n
= a^n(a^3+ka+1) + b^n(b^3+bk+1) + c^n(c^3+kc+1)
= a^n(0)+b^n(0)+c^n(0) since a,b and c are roots
=0
=RHS

Are you asking: When P(x) is divided by (ax+b), remainder=__ , when divided by (cx+d), remainder=__. Hence find the remainder when P(x) is divided by (ax+b)(cx+d). ??
If so, you just generate the equation of P(x) (say it's a monic cubic)

p(x)=(ax+b)Q(x) +___
p(x)=(cx+d)T(x) +___

Both these equations are congruent so you equate. Usually you're given the linear factors so you can sub it into p(x)=(ax+b)(cx+d)Z(x) + mx+n and from there you generate another two equations which you solve simultaneously to find m and n.

 

[braintic]

Multiply the equation by x^n giving x^(n+3) + kx^(n+1)+x^n=0.
This equation still has roots α, β and γ. It also has n roots of zero, which are irrelevant to the question.

Substituting α, β and γ:
α^(n+3) + kα^(n+1) + α^n=0
β^(n+3) + kβ^(n+1) + β^n=0
γ^(n+3) + kγ^(n+1) + γ^n=0

Adding:
[ α^(n+3) + β^(n+3) + γ^(n+3) ] + k[α^(n+1) + β^(n+1) + γ^(n+1)] + [α^n + β^n + γ^n] = 0

ie. Sn+3 + kSn+1 + Sn = 0

Edit: Its essentially the same as the previous answer, but it gives you a method when the answer is not supplied in the question.

 

[Carrotsticks]