Power sum of polynomial roots

Synthetic division can be applied to find the power sums of roots of a polynomial $f(x)$
by dividing $f’(x)$ with $f(x)$

$f(x)=x^2+2x-3=0$ has two real roots $x_1=-3,~x_2=1$
$f’(x)=2x+2$

$\begin{array}{c|cccccccccccc} ~ & 2 & 2 \\ -2 & ~ & -4 & 4 & -20 & 52 & -164 & 484 & -1460 & 4372 & -13124 \\ 3 & ~ & ~ & 6 & -6 & 30 & -78 & 246 & -726 & 2190 & -6558 & 19686 \\ \hline ~ & 2 & -2 & 10 & -26 & 82 & -242 & 730 & -2186 & 6562 & -19682 & 19686 \end{array}$

The coefficients of the quotient show that
$\displaystyle \sum_{k=1}^2 x_k^0=2,~\sum_{k=1}^2 x_k^1=-2,~\sum_{k=1}^2 x_k^2=10,~\sum_{k=1}^2 x_k^3=-26,\dots$

This result is caused by the formula $\dfrac{f'(x)}{f(x)}=\sum_{k=0}^\infty \dfrac{1}{x^{k+1}}\left(\sum_{j=1}^n x_j^k\right)$

Reference: The application of Synthetic division in Polynomial theory (Chinese) Section 1.1.5

Proof

Example

Python program to generate Synthetic division in LaTeX code

dividend=[27,-18,0];#input by user, 27x^2-18x for example
divisor=[9,-9,0,1];#input by user, 9x^3-9x^2+1 for example
 
def roundint(x):
 if x == int(x):
  return str(int(x))
 else:
  for k in range(1,100):
   T=0
   if x*k == int(x*k):
    T=1;break;
  if T==1:
   return "\dfrac{"+str(int(x*k))+"}{"+str(k)+"}"
  else:
   return str(round(x,3))
qlen=11;e=0;
dlen=len(divisor)-1;
ddlen=len(dividend);
dd="~ "
for k in range(0,ddlen):
 dividend[k]=dividend[k]/divisor[0]
 dd=dd+"& "+roundint(dividend[k])+" "
for k in range(0,qlen-ddlen+1):
 dividend.append(0)
dd=dd+"\\\\\n"
for k in range(1,dlen+1):
 divisor[k]=-divisor[k]/divisor[0]
q=[]
q.append(dividend[0])
qq="~ & "+roundint(q[0])+" "
s="$\\begin{array}{c|"
s2=[]
for k in range(1,dlen+1):
 s2.append(roundint(divisor[k])+" ")
for k in range(0,dlen):
 for i in range(0,k+1):
  s2[k]=s2[k]+"& ~ "
for i in range(0,qlen-dlen):
 t=dividend[i+1]
 for k in range(0,dlen):
  tt=q[i]*divisor[k+1]
  s2[k]=s2[k]+"& "+roundint(tt)+" "
  if i-k >= 0:
   t=t+q[i-k]*divisor[k+1]
 q.append(t)
 if t==0:
  e=e+1
 else:
  e=0
 if e>=dlen and i>=ddlen-1:
  break
 qq=qq+"& "+roundint(t)+" "
for i in range(qlen-dlen,qlen-1):
 if e>=dlen and i>=ddlen-1:
  break
 t=dividend[i+1]
 for k in range(0,dlen):
  if i-k < qlen-dlen:
   t=t+q[i-k]*divisor[k+1]
 q.append(t)
 qq=qq+"& "+roundint(t)+" "
for k in range(0,qlen+1):
 s=s+"c"
s=s+"}\n"+dd
for k in range(0,dlen):
 s=s+s2[k]+"\\\\\n"
s=s+"\\hline\n"+qq+"\\end{array}$"
print(s)

dividend=[27,-18,0];#input by user, 27x^2-18x for example
divisor=[9,-9,0,1];#input by user, 9x^3-9x^2+1 for example

def roundint(x):
 if x == int(x):
  return str(int(x))
 else:
  for k in range(1,100):
   T=0
   if x*k == int(x*k):
    T=1;break;
  if T==1:
   return "\dfrac{"+str(int(x*k))+"}{"+str(k)+"}"
  else:
   return str(round(x,3))
qlen=11;e=0;
dlen=len(divisor)-1;
ddlen=len(dividend);
dd="~ "
for k in range(0,ddlen):
 dividend[k]=dividend[k]/divisor[0]
 dd=dd+"& "+roundint(dividend[k])+" "
for k in range(0,qlen-ddlen+1):
 dividend.append(0)
dd=dd+"\\\\\n"
for k in range(1,dlen+1):
 divisor[k]=-divisor[k]/divisor[0]
q=[]
q.append(dividend[0])
qq="~ & "+roundint(q[0])+" "
s="$\\begin{array}{c|"
s2=[]
for k in range(1,dlen+1):
 s2.append(roundint(divisor[k])+" ")
for k in range(0,dlen):
 for i in range(0,k+1):
  s2[k]=s2[k]+"& ~ "
for i in range(0,qlen-dlen):
 t=dividend[i+1]
 for k in range(0,dlen):
  tt=q[i]*divisor[k+1]
  s2[k]=s2[k]+"& "+roundint(tt)+" "
  if i-k >= 0:
   t=t+q[i-k]*divisor[k+1]
 q.append(t)
 if t==0:
  e=e+1
 else:
  e=0
 if e>=dlen and i>=ddlen-1:
  break
 qq=qq+"& "+roundint(t)+" "
for i in range(qlen-dlen,qlen-1):
 if e>=dlen and i>=ddlen-1:
  break
 t=dividend[i+1]
 for k in range(0,dlen):
  if i-k < qlen-dlen:
   t=t+q[i-k]*divisor[k+1]
 q.append(t)
 qq=qq+"& "+roundint(t)+" "
for k in range(0,qlen+1):
 s=s+"c"
s=s+"}\n"+dd
for k in range(0,dlen):
 s=s+s2[k]+"\\\\\n"
s=s+"\\hline\n"+qq+"\\end{array}$"
print(s)

pywindow code

[pywindow]
dividend=[27,-18,0];#input by user, 27x^2-18x for example
divisor=[9,-9,0,1];#input by user, 9x^3-9x^2+1 for example

def roundint(x):
 if x == int(x):
  return str(int(x))
 else:
  for k in range(1,100):
   T=0
   if x*k == int(x*k):
    T=1;break;
  if T==1:
   return "\dfrac{"+str(int(x*k))+"}{"+str(k)+"}"
  else:
   return str(round(x,3))
qlen=11;e=0;
dlen=len(divisor)-1;
ddlen=len(dividend);
dd="~ "
for k in range(0,ddlen):
 dividend[k]=dividend[k]/divisor[0]
 dd=dd+"& "+roundint(dividend[k])+" "
for k in range(0,qlen-ddlen+1):
 dividend.append(0)
dd=dd+"\\\\\n"
for k in range(1,dlen+1):
 divisor[k]=-divisor[k]/divisor[0]
q=[]
q.append(dividend[0])
qq="~ & "+roundint(q[0])+" "
s="$\\begin{array}{c|"
s2=[]
for k in range(1,dlen+1):
 s2.append(roundint(divisor[k])+" ")
for k in range(0,dlen):
 for i in range(0,k+1):
  s2[k]=s2[k]+"& ~ "
for i in range(0,qlen-dlen):
 t=dividend[i+1]
 for k in range(0,dlen):
  tt=q[i]*divisor[k+1]
  s2[k]=s2[k]+"& "+roundint(tt)+" "
  if i-k >= 0:
   t=t+q[i-k]*divisor[k+1]
 q.append(t)
 if t==0:
  e=e+1
 else:
  e=0
 if e>=dlen and i>=ddlen-1:
  break
 qq=qq+"& "+roundint(t)+" "
for i in range(qlen-dlen,qlen-1):
 if e>=dlen and i>=ddlen-1:
  break
 t=dividend[i+1]
 for k in range(0,dlen):
  if i-k < qlen-dlen:
   t=t+q[i-k]*divisor[k+1]
 q.append(t)
 qq=qq+"& "+roundint(t)+" "
for k in range(0,qlen+1):
 s=s+"c"
s=s+"}\n"+dd
for k in range(0,dlen):
 s=s+s2[k]+"\\\\\n"
s=s+"\\hline\n"+qq+"\\end{array}$"
print(s)
[/pywindow]