Find a method by which one can compute the coefficients of $P(x) = x^6 + a_1x^5 + \cdots+  a_6$ from the roots of $P(x) = 0$ by performing not more than $15$ additions and $15$ multiplications.
To find the coefficients of the polynomial \( P(x) = x^6 + a_1x^5 + a_2x^4 + a_3x^3 + a_4x^2 + a_5x + a_6 \) from its roots \( r_1, r_2, r_3, r_4, r_5, r_6 \), we can use Vieta's formulas. Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. Specifically, for a polynomial of degree 6, the coefficients are given by:

\[
\begin{aligned}
a_1 &= -(r_1 + r_2 + r_3 + r_4 + r_5 + r_6), \\
a_2 &= r_1r_2 + r_1r_3 + r_1r_4 + r_1r_5 + r_1r_6 + r_2r_3 + r_2r_4 + r_2r_5 + r_2r_6 + r_3r_4 + r_3r_5 + r_3r_6 + r_4r_5 + r_4r_6 + r_5r_6, \\
a_3 &= -(r_1r_2r_3 + r_1r_2r_4 + r_1r_2r_5 + r_1r_2r_6 + r_1r_3r_4 + r_1r_3r_5 + r_1r_3r_6 + r_1r_4r_5 + r_1r_4r_6 + r_1r_5r_6 \\
&\quad + r_2r_3r_4 + r_2r_3r_5 + r_2r_3r_6 + r_2r_4r_5 + r_2r_4r_6 + r_2r_5r_6 + r_3r_4r_5 + r_3r_4r_6 + r_3r_5r_6 + r_4r_5r_6), \\
a_4 &= r_1r_2r_3r_4 + r_1r_2r_3r_5 + r_1r_2r_3r_6 + r_1r_2r_4r_5 + r_1r_2r_4r_6 + r_1r_2r_5r_6 + r_1r_3r_4r_5 + r_1r_3r_4r_6 + r_1r_3r_5r_6 \\
&\quad + r_1r_4r_5r_6 + r_2r_3r_4r_5 + r_2r_3r_4r_6 + r_2r_3r_5r_6 + r_2r_4r_5r_6 + r_3r_4r_5r_6, \\
a_5 &= -(r_1r_2r_3r_4r_5 + r_1r_2r_3r_4r_6 + r_1r_2r_3r_5r_6 + r_1r_2r_4r_5r_6 + r_1r_3r_4r_5r_6 + r_2r_3r_4r_5r_6), \\
a_6 &= r_1r_2r_3r_4r_5r_6.
\end{aligned}
\]

We can compute these coefficients step-by-step using the given roots and ensuring that we do not exceed 15 additions and 15 multiplications.

1. **Calculate sums and products of pairs of roots:**
   \[
   \begin{aligned}
   a &= r_1 + r_2, \\
   b &= r_1r_2, \\
   c &= r_3 + r_4, \\
   d &= r_3r_4, \\
   e &= r_5 + r_6, \\
   f &= r_5r_6.
   \end{aligned}
   \]

2. **Combine sums and products for groups of four roots:**
   \[
   \begin{aligned}
   g &= a + c = r_1 + r_2 + r_3 + r_4, \\
   h &= eg = r_1r_5 + r_2r_5 + r_3r_5 + r_4r_5 + r_1r_6 + r_2r_6 + r_3r_6 + r_4r_6, \\
   i &= ac = r_1r_3 + r_2r_3 + r_1r_4 + r_2r_4, \\
   j &= b + d = r_1r_2 + r_3r_4, \\
   k &= i + j = r_1r_2 + r_1r_3 + r_1r_4 + r_2r_3 + r_2r_4 + r_3r_4.
   \end{aligned}
   \]

3. **Combine sums and products for all six roots:**
   \[
   \begin{aligned}
   l &= k + h = r_1r_2 + r_1r_3 + r_1r_4 + r_2r_3 + r_2r_4 + r_3r_4 + r_1r_5 + r_2r_5 + r_3r_5 + r_4r_5 + r_1r_6 + r_2r_6 + r_3r_6 + r_4r_6, \\
   m &= ke = r_1r_2r_5 + r_1r_3r_5 + r_1r_4r_5 + r_2r_3r_5 + r_2r_4r_5 + r_3r_4r_5 + r_1r_2r_6 + r_1r_3r_6 + r_1r_4r_6 + r_2r_3r_6 + r_2r_4r_6 + r_3r_4r_6, \\
   n &= ad = r_1r_3r_4 + r_2r_3r_4, \\
   o &= bc = r_1r_2r_3 + r_1r_2r_4, \\
   p &= n + o = r_1r_2r_3 + r_1r_2r_4 + r_1r_3r_4 + r_2r_3r_4, \\
   q &= gf = r_1r_5r_6 + r_2r_5r_6 + r_3r_5r_6 + r_4r_5r_6, \\
   r &= m + q = r_1r_2r_5 + r_1r_3r_5 + r_1r_4r_5 + r_2r_3r_5 + r_2r_4r_5 + r_3r_4r_5 + r_1r_2r_6 + r_1r_3r_6 + r_1r_4r_6 + r_2r_3r_6 + r_2r_4r_6 + r_3r_4r_6 + r_1r_5r_6 + r_2r_5r_6 + r_3r_5r_6 + r_4r_5r_6.
   \end{aligned}
   \]

4. **Calculate the coefficients:**
   \[
   \begin{aligned}
   a_1 &= -(g + e) = -(r_1 + r_2 + r_3 + r_4 + r_5 + r_6), \\
   a_2 &= l + f = r_1r_2 + r_1r_3 + r_1r_4 + r_1r_5 + r_1r_6 + r_2r_3 + r_2r_4 + r_2r_5 + r_2r_6 + r_3r_4 + r_3r_5 + r_3r_6 + r_4r_5 + r_4r_6 + r_5r_6, \\
   a_3 &= -(r + p) = -(r_1r_2r_3 + r_1r_2r_4 + r_1r_2r_5 + r_1r_2r_6 + r_1r_3r_4 + r_1r_3r_5 + r_1r_3r_6 + r_1r_4r_5 + r_1r_4r_6 + r_1r_5r_6 + r_2r_3r_4 + r_2r_3r_5 + r_2r_3r_6 + r_2r_4r_5 + r_2r_4r_6 + r_2r_5r_6 + r_3r_4r_5 + r_3r_4r_6 + r_3r_5r_6 + r_4r_5r_6), \\
   a_4 &= v + s = r_1r_2r_3r_4 + r_1r_2r_3r_5 + r_1r_2r_3r_6 + r_1r_2r_4r_5 + r_1r_2r_4r_6 + r_1r_2r_5r_6 + r_1r_3r_4r_5 + r_1r_3r_4r_6 + r_1r_3r_5r_6 + r_1r_4r_5r_6 + r_2r_3r_4r_5 + r_2r_3r_4r_6 + r_2r_3r_5r_6 + r_2r_4r_5r_6 + r_3r_4r_5r_6, \\
   a_5 &= -(w + x) = -(r_1r_2r_3r_4r_5 + r_1r_2r_3r_4r_6 + r_1r_2r_3r_5r_6 + r_1r_2r_4r_5r_6 + r_1r_3r_4r_5r_6 + r_2r_3r_4r_5r_6), \\
   a_6 &= sf = r_1r_2r_3r_4r_5r_6.
   \end{aligned}
   \]

The final answer is the coefficients \( \boxed{ a_1, a_2, a_3, a_4, a_5, a_6 } \) as derived above.