Product with polynomial roots

by fungarwai, Mar 18, 2024, 11:35 AM

Vandermonde polynomial type

$\displaystyle \prod_{1\le i<j\le m} (x_j-x_i)^2
=(-1)^{\binom{m}{2}}\prod_{j=1}^m f'(x_j),
~f(x)=\prod_{i=1}^m (x-x_i)$

Proof

Example

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 $f(x)$
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

  1. dividend=[27,-18,0];#input by user, 27x^2-18x for example
  2. divisor=[9,-9,0,1];#input by user, 9x^3-9x^2+1 for example
  3.  
  4. def roundint(x):
  5. if x == int(x):
  6. return str(int(x))
  7. else:
  8. for k in range(1,100):
  9. T=0
  10. if x*k == int(x*k):
  11. T=1;break;
  12. if T==1:
  13. return "\dfrac{"+str(int(x*k))+"}{"+str(k)+"}"
  14. else:
  15. return str(round(x,3))
  16. qlen=11;e=0;
  17. dlen=len(divisor)-1;
  18. ddlen=len(dividend);
  19. dd="~ "
  20. for k in range(0,ddlen):
  21. dividend[k]=dividend[k]/divisor[0]
  22. dd=dd+"& "+roundint(dividend[k])+" "
  23. for k in range(0,qlen-ddlen+1):
  24. dividend.append(0)
  25. dd=dd+"\\\\\n"
  26. for k in range(1,dlen+1):
  27. divisor[k]=-divisor[k]/divisor[0]
  28. q=[]
  29. q.append(dividend[0])
  30. qq="~ & "+roundint(q[0])+" "
  31. s="$\\begin{array}{c|"
  32. s2=[]
  33. for k in range(1,dlen+1):
  34. s2.append(roundint(divisor[k])+" ")
  35. for k in range(0,dlen):
  36. for i in range(0,k+1):
  37. s2[k]=s2[k]+"& ~ "
  38. for i in range(0,qlen-dlen):
  39. t=dividend[i+1]
  40. for k in range(0,dlen):
  41. tt=q[i]*divisor[k+1]
  42. s2[k]=s2[k]+"& "+roundint(tt)+" "
  43. if i-k >= 0:
  44. t=t+q[i-k]*divisor[k+1]
  45. q.append(t)
  46. if t==0:
  47. e=e+1
  48. else:
  49. e=0
  50. if e>=dlen and i>=ddlen-1:
  51. break
  52. qq=qq+"& "+roundint(t)+" "
  53. for i in range(qlen-dlen,qlen-1):
  54. if e>=dlen and i>=ddlen-1:
  55. break
  56. t=dividend[i+1]
  57. for k in range(0,dlen):
  58. if i-k < qlen-dlen:
  59. t=t+q[i-k]*divisor[k+1]
  60. q.append(t)
  61. qq=qq+"& "+roundint(t)+" "
  62. for k in range(0,qlen+1):
  63. s=s+"c"
  64. s=s+"}\n"+dd
  65. for k in range(0,dlen):
  66. s=s+s2[k]+"\\\\\n"
  67. s=s+"\\hline\n"+qq+"\\end{array}$"
  68. print(s)


pywindow code
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

$\sum_{i=1}^m \frac{1}{x-x_i}=\frac{f'(x)}{f(x)},~f(x)=\prod_{i=1}^m (x-x_i)$

Credit to the idea posted by ccmmjj in mathchina forum

Proof

$\sum_{i=1}^m \frac{-1}{(x-x_i)^2}=\frac{f''(x)f(x)-[f'(x)]^2}{[f(x)]^2}$

$\sum_{i=1}^m \frac{2}{(x-x_i)^3}=\frac{f'''(x)[f(x)]^2-3f''(x)f'(x)f(x)+2[f'(x)]^3}{[f(x)]^3}$

Identity of a conditional symmetric polynomial

Let $\displaystyle S_{n,m}=\sum_{r_1+r_2+\cdots+r_m=n} x_1^{r_1}x_2^{r_2}\cdots x_m^{r_m},~\sigma_{k,m} =\sum_{1\le j_1 < j_2 < \cdots < j_k \le m} x_{j_1} \dots x_{j_k}$
$\displaystyle \boxed{S_{j,m}=\sum_{k=1}^{m-1} (-1)^{k-1} \sigma_{k,m} S_{j-k,m}+(-1)^{m-1}\sigma_{m,m} S_{j-m,m}}$
$\displaystyle \text{or}~ \boxed{S_{j,m}=\sum_{k=1}^{j-1} (-1)^{k-1} \sigma_{k,m} S_{j-k,m}+(-1)^{j-1}\sigma_{j,m}}$

Proof

Example

$\displaystyle S_{j,K}=\sum_{k=1}^{M-1} (-1)^{k-1}
\sigma_{k,M} S_{j-k,K}+(-1)^{M-1}\sigma_{M,M}S_{j-M,K},~\forall K\le M\le m$

Proof

Example

Lagrange polynomial type

$S_{n,m}=\sum_{i=1}^m x_i^n \prod_{j=1\atop i\neq j}^m\frac{1}{x_i-x_j}
=\begin{cases}0 & 0\le n\le m-2\\1 & n=m-1\end{cases}$

Proof

$S_{n,m}=\sum_{i=1}^m x_i^n \prod_{j=1\atop i\neq j}^m\frac{1}{x_i-x_j}
=\begin{cases}0 & 0\le n\le m-2\\1 & n=m-1\\
\displaystyle
\boxed{\sum_{r_1+r_2+\cdots+r_m=n-m+1} x_1^{r_1}x_2^{r_2}\cdots x_m^{r_m}}
 & n\ge m
\end{cases}$

$S_{m,m}=\sum a$
$S_{m+1,m}=\sum a^2+\sum ab$
$S_{m+2,m}=\sum a^3+\sum a^2 b+\sum abc$
$S_{m+3,m}=\sum a^4+\sum a^3 b+\sum a^2 b^2+\sum a^2 bc+\sum abcd$

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

$S_{n,m}=\sum_{i=1}^m x_i^n \prod_{j=1\atop i\neq j}^m\frac{1}{x_i-x_j}=\begin{cases}0 & 0\le n\le m-2\\1 & n=m-1\\
\displaystyle
\boxed{\sum_{r_1 + 2r_2 + \cdots + mr_m = s \atop r_1\ge 0, \ldots, r_m\ge 0} (-1)^n \frac{(r_1 + r_2 + \cdots + r_m)!}{r_1!r_2! \cdots r_m!} \prod_{i=1}^m (-\sigma_{i,m})^{r_i}} & n=s+m-1\ge m
\end{cases}$

where $\sigma_{k,m} =\sigma_k (x_1 , \ldots , x_m )=\sum_{1\le  j_1 < j_2 < \cdots < j_k \le m} x_{j_1} \dots x_{j_k}$

$S_{m,m}=\sigma_{1,m}$
$S_{m+1,m}=\sigma_{1,m}^2-\sigma_{2,m}$
$S_{m+2,m}=\sigma_{1,m}^3-2\sigma_{1,m}\sigma_{2,m}+\sigma_{3,m}$
$S_{m+3,m}=\sigma_{1,m}^4-3\sigma_{1,m}^2\sigma_{2,m}+\sigma_{2,m}^2-2\sigma_{1,m}\sigma_{3,m}+\sigma_{4,m}$

Proof

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 $P(G_{m,n},t)$

Chromatic polynomial of a 2×n grid has been found and named as ladder graph

$P(G_{2,n},t)=t(t-1)(t^2-3t+3)^{n-1}$

My recurrence equation found for Chromatic polynomial of a 3×n grid is as follow

$P(G_{3,n},t)=(t^3-5t^2+11t-10)P(G_{3,n-1},t)-(t^4-7t^3+19t^2-24t+11)P(G_{3,n-2},t)$
$~=(t-2)(t^2-3t+5)P(G_{3,n-1},t)-(t-1)(t^3-6t^2+13t-11)P(G_{3,n-2},t)$

$P(G_{3,1},t)=t(t-1)^2,~P(G_{3,2},t)=t(t-1)(t^2-3t+3)^2$
$P(G_{3,3},t)=(t^3-5t^2+11t-10)t(t-1)(t^2-3t+3)^2
-(t^4-7t^3+19t^2-24t+11)t(t-1)^2$
$~=t(t-1)(t^7-11t^6+55t^5-161t^4+298t^3-350t^2+244t-79)$

[asy]
unitsize(5mm);
int i, j, k;
k=0;
for(i=0; i<k+7; i=i+1){
if(i!=2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
draw((0,j)--(k+6,j));
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$=$", (7,1));
k=8;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+7; i=i+1){
if(i!=10){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (15,1));
k=16;
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
draw((k,j)--(k+5,j));
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
dot((k+6,0.5));dot((k+6,1.5));
draw((k+5,0)--(k+6,0.5));
draw((k+5,1)--(k+6,0.5));
draw((k+5,1)--(k+6,1.5));
draw((k+5,2)--(k+6,1.5));
[/asy]
[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (0,1));label("$=$", (15,1));
k=16;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
draw((k+6,1)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+6,0));dot((k+6,1));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (23,1));
k=24;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
draw((k+6,0)--(k+6,2));dot((k+6,0));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (31,1));
k=32;
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
draw((k,j)--(k+5,j));
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
dot((k+6,0.5));dot((k+6,1.5));
draw((k+5,0)--(k+6,0.5));
draw((k+5,1)--(k+6,0.5));
draw((k+5,1)--(k+6,1.5));
draw((k+5,2)--(k+6,1.5));
[/asy]
[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (0,1));label("$=[(t-1)^3-(t-2)^2]$", (17,1));
k=22;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (28,1));
k=29;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
draw((k+6,0)--(k+6,2));dot((k+6,0));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]
[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (0,1));label("$=(t^3-4t^2+7t-5)$", (17,1));
k=22;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (28,1));
k=29;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
draw((k+6,0)--(k+6,2));dot((k+6,0));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy][asy]
unitsize(5mm);
int i, j, k;
k=0;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
draw((k+6,0)--(k+6,2));dot((k+6,0));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$=$", (7,1));
k=8;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+6,0));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (15,1));
k=16;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
draw((k+5,2)--(k+6,0));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+6,0));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy][asy]
unitsize(5mm);
int i, j, k;
label("$~$", (-7,1));label("$=$", (7,1));
k=8;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+6,0));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (15,1));
k=16;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+6,0));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$+$", (23,1));
k=24;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
draw((k+5,0)..(k+6,1)..(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]
[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (-14,1));label("$=[(t-1)^2-(t-1)]$", (3,1));
k=8;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$+$", (15,1));
k=16;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
draw((k+5,0)..(k+6,1)..(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]
[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (-11,1));label("$=(t-1)(t-2)$", (3,1));
k=7;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$+$", (14,1));
k=15;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
draw((k+5,0)..(k+6,1)..(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]
[asy]
unitsize(5mm);
int i, j, k;
k=0;
for(i=0; i<k+7; i=i+1){
if(i!=2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
draw((0,j)--(k+6,j));
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$=[(t-1)^3-(t-2)^2]$", (11,1));
k=16;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (22,1));
k=23;
draw((k,0)--(k+6,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+6,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
draw((k+6,0)--(k+6,2));dot((k+6,0));dot((k+6,2));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]
[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (-10,1));label("$=[(t-1)^3-(t-2)^2-(t-1)^2+(t-1)]$", (11,1));
k=20;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (26,1));
k=27;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
draw((k+5,0)..(k+6,1)..(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]
[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (0,1));label("$=(t^3-5t^2+10t-7)$", (11,1));
k=16;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (22,1));
k=23;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
draw((k+5,0)..(k+6,1)..(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]
[asy]
unitsize(5mm);
int i, j, k;
k=0;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
draw((k+5,0)..(k+6,1)..(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$=$", (7,1));
k=8;
draw((k,0)--(k+5,0));
draw((k,1)--(k+5,1));
draw((k,2)--(k+5,2));
for(i=k; i<k+6; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (15,1));
k=16;
draw((k,0)--(k+4,0));
draw((k,1)--(k+6,1));
draw((k,2)--(k+4,2));
draw((k+4,2)--(k+6,1));
draw((k+4,0)--(k+6,1));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+5,1));dot((k+6,1));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]
[asy]
unitsize(5mm);
int i, j, k;
k=0;
draw((k,0)--(k+4,0));draw((k+4,0)--(k+6,1));
draw((k,1)--(k+6,1));
draw((k,2)--(k+4,2));draw((k+4,2)--(k+6,1));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+5,1));dot((k+6,1));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$=$", (7,1));
k=8;
draw((k,0)--(k+4,0));draw((k+4,0)--(k+6,1));
draw((k,1)--(k+4,1));draw((k+5,1)--(k+6,1));
draw((k,2)--(k+4,2));draw((k+4,2)--(k+6,1));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+5,1));dot((k+6,1));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-$", (15,1));
k=16;
draw((k,0)--(k+4,0));draw((k+4,0)--(k+6,1));
draw((k,1)--(k+6,1));
draw((k,2)--(k+4,2));draw((k+4,2)--(k+6,1));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+6,1));
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]
[asy]
unitsize(5mm);int i, j, k;
label("$=$", (7,1));
k=8;draw((k,0)--(k+4,0));draw((k+4,0)--(k+6,1));
draw((k,1)--(k+4,1));draw((k+5,1)--(k+6,1));
draw((k,2)--(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+5,1));dot((k+6,1));
label("...", (k+2,0.5));label("...", (k+2,1.5));
label("$-$", (15,1));
k=16;draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));draw((k+4,2)--(k+5,1));
draw((k,2)--(k+4,2));
draw((k+4,0)..(k+6,1)..(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+5,1));
label("...", (k+2,0.5));label("...", (k+2,1.5));
label("$-$", (23,1));
k=24;draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));draw((k,2)--(k+4,2));
draw((k+4,2)--(k+6,1));draw((k+4,0)--(k+6,1));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+6,1));
label("...", (k+2,0.5));label("...", (k+2,1.5));
label("$+$", (31,1));
k=32;draw((k,0)--(k+4,0));draw((k,1)--(k+4,1));draw((k,2)--(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));label("...", (k+2,1.5));
[/asy]
[asy]
unitsize(5mm);int i, j, k;
label("$=$", (7,1));
k=8;draw((k,0)--(k+4,0));draw((k+4,0)--(k+6,1));
draw((k,1)--(k+4,1));draw((k+5,1)--(k+6,1));
draw((k,2)--(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
dot((k+5,1));dot((k+6,1));
label("...", (k+2,0.5));label("...", (k+2,1.5));
label("$-$", (15,1));
k=16;draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));draw((k+4,2)--(k+5,1));
draw((k,2)--(k+4,2));
draw((k+4,0)..(k+6,1)..(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}dot((k+5,1));
label("...", (k+2,0.5));label("...", (k+2,1.5));label("$-$", (23,1));
k=24;draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));draw((k,2)--(k+4,2));
draw((k+4,0)--(k+6,1));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){if(i!=k+2){dot((i,j));}}}
dot((k+6,1));label("...", (k+2,0.5));label("...", (k+2,1.5));label("$+$", (31,1));
k=32;draw((k,0)--(k+4,0));draw((k,1)--(k+4,1));draw((k,2)--(k+4,2));
draw((k+4,0)..(k+5,1)..(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));label("...", (k+2,1.5));
label("$+$", (39,1));
k=40;draw((k,0)--(k+4,0));draw((k,1)--(k+4,1));draw((k,2)--(k+4,2));
for(i=k; i<k+5; i=i+1){if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));label("...", (k+2,1.5));
[/asy]


[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (-14,1));label("$=[(t-1)^2-(t-1)+1]$", (3,1));
k=8;
draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));
draw((k,2)--(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-(t-2)$", (14,1));
k=16;
draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));
draw((k,2)--(k+4,2));
draw((k+4,0)..(k+5,1)..(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]
[asy]
unitsize(5mm);
int i, j, k;
label("$~$", (-10,1));label("$=(t^2-3t+3)$", (3,1));
k=6;
draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));
draw((k,2)--(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
label("$-(t-2)$", (12,1));
k=14;
draw((k,0)--(k+4,0));
draw((k,1)--(k+4,1));
draw((k,2)--(k+4,2));
draw((k+4,0)..(k+5,1)..(k+4,2));
for(i=k; i<k+5; i=i+1){
if(i!=k+2){draw((i,0)--(i,2));}
for(j=0; j<3; j=j+1){
if(i!=k+2){dot((i,j));}}}
label("...", (k+2,0.5));
label("...", (k+2,1.5));
[/asy]
$a_n=(t^3-5t^2+10t-7)a_{n-1}-b_{n-1}$
$b_n=a_n-(t^2-3t+3)a_{n-1}+(t-2)b_{n-1}$

$b_{n-1}=(t^3-5t^2+10t-7)a_{n-1}-a_n$
$b_n=(t^3-5t^2+10t-7)a_n-a_{n+1}$
$(t^3-5t^2+10t-7)a_n-a_{n+1}=a_n-(t^2-3t+3)a_{n-1}+(t-2)[(t^3-5t^2+10t-7)a_{n-1}-a_n]$
$a_{n+1}=(t^3-5t^2+11t-10)a_n-(t^4-7t^3+19t^2-24t+11)a_{n-1}$

[asy]
unitsize(5mm);int i, j, k;
label("$a_1:$", (0,1));
k=2;
for(i=k; i<k+1; i=i+1){
draw((i,0)--(i,2));
for(j=0; j<3; j=j+1){dot((i,j));}}
label("$a_2:$", (5,1));
k=7;
for(i=k; i<k+2; i=i+1){
draw((i,0)--(i,2));
for(j=0; j<3; j=j+1){
draw((k,j)--(k+1,j));
dot((i,j));}}
label("$a_3:$", (11,1));
k=13;
for(i=k; i<k+3; i=i+1){
draw((i,0)--(i,2));
for(j=0; j<3; j=j+1){
draw((k,j)--(k+2,j));
dot((i,j));}}
[/asy]

$a_1=t(t-1)^2,~a_2=t(t-1)(t^2-3t+3)^2$
$a_3=(t^3-5t^2+11t-10)t(t-1)(t^2-3t+3)^2-(t^4-7t^3+19t^2-24t+11)t(t-1)^2$
$~=t(t-1)(t^7+11t^6+55t^5-161t^4+298t^3-350t^2+244t-79)$
This post has been edited 17 times. Last edited by fungarwai, Aug 27, 2021, 12:50 PM

Notable algebra methods with proofs and examples

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fungarwai
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  • Nice blog!

    by Inconsistent, Mar 18, 2024, 2:41 PM

  • hey, nice blog! really enjoyed the content here and thank you for this contribution to aops. Sure to subscribe! :)

    by thedodecagon, Jan 22, 2022, 1:33 AM

  • thanks for this

    by jasperE3, Dec 3, 2021, 10:01 PM

  • I am working as accountant and studying as ACCA student now.
    I graduated applied mathematics at bachelor degree in Jinan University but I still have no idea to find a specific job with this..

    by fungarwai, Aug 28, 2021, 4:54 AM

  • Awesome algebra blog :)

    by Euler1728, Mar 22, 2021, 5:37 AM

  • I wonder if accountants need that kind of math tho

    by Hamroldt, Jan 14, 2021, 10:55 AM

  • Nice!!!!

    by Delta0001, Dec 12, 2020, 10:20 AM

  • this is very interesting i really appericate it :)

    by vsamc, Oct 29, 2020, 4:42 PM

  • this is god level

    by Hamroldt, Sep 4, 2020, 7:48 AM

  • Super Blog! You are Pr0! :)

    by Functional_equation, Aug 23, 2020, 7:43 AM

  • Great blog!

    by freeman66, May 31, 2020, 5:40 AM

  • cool thx! :D

    by erincutin, May 18, 2020, 4:55 PM

  • How so op???

    by Williamgolly, Apr 30, 2020, 2:42 PM

  • Beautiful

    by Al3jandro0000, Apr 25, 2020, 3:11 AM

  • Nice method :)

    by Feridimo, Jan 23, 2020, 5:05 PM

  • This is nice!

    by mufree, May 26, 2019, 6:40 AM

  • Wow! So much Algebra.

    by AnArtist, Mar 15, 2019, 1:19 PM

  • :omighty: :omighty:

    by AlastorMoody, Feb 9, 2019, 5:17 PM

  • 31415926535897932384626433832795

    by lkarhat, Dec 25, 2018, 11:53 PM

  • rip 0 shouts and 0 comments until now

    by harry1234, Nov 17, 2018, 8:56 PM

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