Sum of fifth power of roots

Question

What is the sum of fifth power of roots of the equation $$x^3+qx+r$$.I tried expanding $$(a+b+c)^5$$ but it didn't work instead it is becoming more and more complex.

Answer 1

$$x^3+qx+r=0\implies a+b+c=0, ab+bc+ca=q,abc=-r$$

$$\implies \sum a^2=(a+b+c)^2-2(ab+bc+ca)$$

and multiplying the given eqaution by $x^n$ $$ x^{n+3}+qx^{n+1}+rx^n=0$$

$$\implies a^{n+3}+qa^{n+1}+ra^n=0$$

$$\implies\sum a^{n+3}+q\sum a^{n+1}+r\sum a^n=0$$

$$n=0\implies\sum a^3+q\sum a+r\sum 1=0\iff \sum a^3=-q\cdot0-r\cdot3$$

$$n=2\implies\sum a^5+q\sum a^3+r\sum a^2=0\iff \sum a^5=-q\sum a^3-r\sum a^2$$

Reference : Newton's Sums

Answer 2

$$\underbrace{x^3+qx+r=0}_{a,b,c}$$

By Viete's relations,

$$a+b+c=0$$

$$ab+bc+ca=q$$

$$abc=-r$$

Since roots satisfy the equation, we have, $$a^3+qa+r=0$$

$$b^3+qb+r=0$$ $$c^3+qc+r=0$$

Or equivalently,

$$a^{n+3}+qa^{n+1}+ra^{n}=0\\\vdots$$

If we denote $S_n=a^n+b^n+c^n$,adding these three equations, we have,

$$S_{n+3}+qS_{n+1}+rS_n=0\tag{1}$$

$$S_5=-qS_3-rS_2 = -q(-qS_1-rS_0)-rS_2$$

We have $S_1=0$ and $S_2=(a+b+c)^2-2(ab+bc+ca)=-2q$

$$S_5=-q(0-3r)-r(-2q)=5qr$$