About the roots of a quadratic equation

Question

Let $m_1$ and $m_2$ the real and diferent roots of the quadratic equation $ax^2+bx+c=0$. Do you know some way to write $m_1^k + m_2^k$ in a simplest form (linear, for example) using just $a,b,c,m_1$ and $m_2$?

Thanks for the attention!

Answer 1

You can use Newton's polynomial to calculate any value of $m_1^k + m_2^k$, without actually calculating the roots and using just the coefficients in front of the variable. Actually it works for any degree polynomial. Let $s_k = m_1^k + m_2^k$ and $m_1$ and $m_2$ be roots of $ax^2 + bx + c=0$. Then we have:

$$as_1 + b = 0$$ $$as_2 + bs_1 + 2c= 0$$ $$as_3 + bs_2 + cs_1 = 0$$ $$as_4 + bs_3 + cs_2 = 0$$ $$ \ldots$$

You can find more about the general formula and Newton's polynomials here.

This maybe isn't what you are, assuming that you want just linear expression, but it's recursive one and should satisfy your requirements