The application of Synthetic division in Polynomial theory
by fungarwai, Feb 9, 2023, 4:40 AM
Power sum of polynomial roots
Synthetic division can be applied to find the power sums of roots of a polynomial
by dividing with
has two real roots
The coefficients of the quotient show that
This result is caused by the formula
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
pywindow code
Synthetic division can be applied to find the power sums of roots of a polynomial
by dividing with
has two real roots
The coefficients of the quotient show that
This result is caused by the formula
Reference: The application of Synthetic division in Polynomial theory (Chinese) Section 1.1.5
Proof
Let ,
Suppose , the Laurent series of can be expressed as
Each term in the Laurent series is unique.
As the coefficients of the quotient are determined one by one with Synthetic division, the power sums of polynomial roots are computed.
Suppose , the Laurent series of can be expressed as
Each term in the Laurent series is unique.
As the coefficients of the quotient are determined one by one with Synthetic division, the power sums of polynomial roots are computed.
Example
a,b,c are roots of the equation x³-9x+9 = 0.
Find the value of -
1/a³ + 1/b³ + 1/c³
1/a⁵ + 1/b⁵ + 1/c⁵.
Reference: Sum of powers of roots
Let
Compute with Synthetic division
i.e.
Find the value of -
1/a³ + 1/b³ + 1/c³
1/a⁵ + 1/b⁵ + 1/c⁵.
Reference: Sum of powers of roots
Let
Compute with Synthetic division
i.e.
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)
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]
This post has been edited 1 time. Last edited by fungarwai, Feb 9, 2023, 4:57 AM
Summation with polynomial roots
by fungarwai, May 26, 2020, 12:52 PM
Reciprocal type
Credit to the idea posted by ccmmjj in mathchina forum
Proof
Identity of a conditional symmetric polynomial
Let
Proof
Example
Proof
Suppose is true for
Example
Lagrange polynomial type
Proof
Best Proof Credit to the idea with four variables posted by natmath at #2
Proof Credit to the idea with three variables posted by gasgous at #10
Other Proof
where
Proof
Example
Credit to the idea posted by ccmmjj in mathchina forum
Proof
Identity of a conditional symmetric polynomial
Let
Proof
Distribute balls into boxes
Each box has balls
Let be the event of box contains at least one ball
By Inclusion–exclusion principle
satisfies for the number of any combinations of , therefore
Each box has balls
Let be the event of box contains at least one ball
By Inclusion–exclusion principle
satisfies for the number of any combinations of , therefore
Example
For the combination as example
which is the same as the situation of the coefficient of
Other examples
which is the same as the situation of the coefficient of
Other examples
Proof
Suppose is true for
Example
Lagrange polynomial type
Proof
Best Proof Credit to the idea with four variables posted by natmath at #2
Proof Credit to the idea with three variables posted by gasgous at #10
Other Proof
Let
such that
where
The coefficient of should be 0
i.e.
And then,
Let
As
Credit to the proof posted by natmath at #4
such that
where
The coefficient of should be 0
i.e.
And then,
Let
As
Credit to the proof posted by natmath at #4
https://artofproblemsolving.com/community/c4h2901132
Consider the polynomial
The desired expression is the coefficient of of . We know that (and cyclically) so
But is a quadratic so the coefficient of and should be . This means that So
The coefficient of is .
Consider the polynomial
The desired expression is the coefficient of of . We know that (and cyclically) so
But is a quadratic so the coefficient of and should be . This means that So
The coefficient of is .
where
Proof
When ,
When ,
Suppose
When ,
Suppose
Example
This post has been edited 23 times. Last edited by fungarwai, May 19, 2023, 12:07 PM
Chromatic polynomial of a 3×n grid
by fungarwai, Apr 26, 2020, 1:26 AM
Denote Chromatic polynomial of a m×n grid as
Chromatic polynomial of a 2×n grid has been found and named as ladder graph
My recurrence equation found for Chromatic polynomial of a 3×n grid is as follow
Chromatic polynomial of a 2×n grid has been found and named as ladder graph
My recurrence equation found for Chromatic polynomial of a 3×n grid is as follow
This post has been edited 17 times. Last edited by fungarwai, Aug 27, 2021, 12:50 PM
Notable algebra methods with proofs and examples
fungarwaiArchives
March 2024
February 2022
December 2021
September 2020
May 2020
April 2020
September 2019
January 2019
December 2018
Shouts
Submit
20 shouts
Tags
About Owner
- Posts: 859
- Joined: Mar 11, 2017
Blog Stats
- Blog created: Sep 15, 2018
- Total entries: 18
- Total visits: 5839
- Total comments: 8
Search Blog