Translating Large x-Values Before Curve Fitting

Find a polynomial that fits the points

                       (2005, 1)    (2006, 2)   (2007,3)   (2009,4)   (2010,10)

Because the given x-values are large, use the translation z = x - 2007 to obtain
   
                                 (-2, 1)   (-1, 2)   (0, 3)   (2, 4)   (3, 10)

This makes the polynomial functions as

p(-2) = a₀ + a₁(-2) + a₂(-2)² + a₃(-2)³ + a₄(-2)⁴ = a₀ - 2a₁ + 4a₂ – 8a₃ + 16a₄ = 1

p(-1) = a₀ + a₁(-1) + a₂(-1)² + a₃(-1)³ + a₄(-1)⁴ = a₀ - a₁ + a₂ –  a₃ + a₄ = 2

p(0) = a₀ + a₁(0) + a₂(0)² + a₃(0)³ + a₄(0)⁴ = a₀ = 3
p(2) = a₀ + a₁(2) + a₂(2)² + a₃(2)³ + a₄(2)⁴ = a₀ + 2a₁ + 4a₂ + 8a₃ + 16a₄ = 4
p(3) = a₀ + a₁(3) + a₂(3)² + a₃(3)³ + a₄(3)⁴ = a₀ + 3a₁ + 9a₂ + 27a₃ + 81a₄ = 10

Then use the Gauss-Jordan Elimination. At the end, the solution is a₀=3, a₁=29/60, a₂=-67/120, a₃=1/15. and a₄=13/120. This means that p(x) = 3 + 29/60x - 67/120x² + 1/15x³ + 13/120x⁴. But because we used z to translate the x-value, the p(x) is now written as 3 + 29/60(x-2007) - 67/120(x-2007)² + 1/15(x-2007)³ + 13/120(x-2007)⁴

  
                                                                              
                                                           

1	-2	4	-8	16	1
1	-1	1	-1	1	2
1	0	0	0	0	3
1	2	4	8	16	4
1	3	9	27	81	10

    
                       

-1(R1) + R2 ---> R2

1	-2	4	-8	16	1
0	1	-3	7	-15	1
1	0	0	0	0	3
1	2	4	8	16	4
1	3	9	27	81	10

-1(R1) + R3 ---> R3

1	-2	4	-8	16	1
0	1	-3	7	-15	1
0	2	-4	8	-16	2
1	2	4	8	16	4
1	3	9	27	81	10

-1(R1) + R4 ---> R4

1	-2	4	-8	16	1
0	1	-3	7	-15	1
0	2	-4	8	-16	2
0	4	0	16	0	3
1	3	9	27	81	10

-1(R1) + R5 ---> R5

1	-2	4	-8	16	1
0	1	-3	7	-15	1
0	2	-4	8	-16	2
0	4	0	16	0	3
0	5	5	35	65	9

-2(R2) + R3 ---> R3

1	-2	4	-8	16	1
0	1	-3	7	-15	1
0	0	2	-6	14	0
0	4	0	16	0	3
0	5	5	35	65	9

-4(R2) + R4 ---> R4

1	-2	4	-8	16	1
0	1	-3	7	-15	1
0	0	2	-6	14	0
0	0	12	-12	60	-1
0	5	5	35	65	9

-5(R2) + R5 ---> R5

1	-2	4	-8	16	1
0	1	-3	7	-15	1
0	0	2	-6	14	0
0	0	12	-12	60	-1
0	0	20	0	140	4

1/2(R3) ---> R3

1	-2	4	-8	16	1
0	1	-3	7	-15	1
0	0	1	-3	7	0
0	0	12	-12	60	-1
0	0	20	0	140	4

-12(R3) + R4 ---> R4

1	-2	4	-8	16	1
0	1	-3	7	-15	1
0	0	1	-3	7	0
0	0	0	24	-24	-1
0	0	20	0	140	4

-20(R3) + R5 ---> R5

1	-2	4	-8	16	1
0	1	-3	7	-15	1
0	0	1	-3	7	0
0	0	0	24	-24	-1
0	0	0	60	0	4

1/24(R4)

1	-2	4	-8	16	1
0	1	-3	7	-15	1
0	0	1	-3	7	0
0	0	0	1	-1	-1/24
0	0	0	60	0	4

-60(R4) + R5 ---> R5

1	-2	4	-8	16	1
0	1	-3	7	-15	1
0	0	1	-3	7	0
0	0	0	1	-1	-1/24
0	0	0	0	60	13/2

1/60(R5) ---> R5

1	-2	4	-8	16	1
0	1	-3	7	-15	1
0	0	1	-3	7	0
0	0	0	1	-1	-1/24
0	0	0	0	1	13/120

1(R5) + R4 ---> R4

1	-2	4	-8	16	1
0	1	-3	7	-15	1
0	0	1	-3	7	0
0	0	0	1	0	1/15
0	0	0	0	1	13/120

-7(R5) + R3 ---> R3

1	-2	4	-8	16	1
0	1	-3	7	-15	1
0	0	1	-3	0	-91/120
0	0	0	1	0	1/15
0	0	0	0	1	13/120

15(R5) + R2 ---> R2

1	-2	4	-8	16	1
0	1	-3	7	0	21/8
0	0	1	-3	0	-91/120
0	0	0	1	0	1/15
0	0	0	0	1	13/120

-16(R5) + R1 ---> R1

1	-2	4	-8	0	-11/15
0	1	-3	7	0	21/8
0	0	1	-3	0	-91/120
0	0	0	1	0	1/15
0	0	0	0	1	13/120

3(R4) + R3 ---> R3

1	-2	4	-8	0	-11/15
0	1	-3	7	0	21/8
0	0	1	0	0	-67/120
0	0	0	1	0	1/15
0	0	0	0	1	13/120

-7(R4) + R2 ---> R2

1	-2	4	-8	0	-11/15
0	1	-3	0	0	259/120
0	0	1	0	0	-67/120
0	0	0	1	0	1/15
0	0	0	0	1	13/120

8(R4) + R1 ---> R1

1	-2	4	0	0	-1/5
0	1	-3	0	0	259/120
0	0	1	0	0	-67/120
0	0	0	1	0	1/15
0	0	0	0	1	13/120

3(R3) + R2 ---> R2

1	-2	4	0	0	-1/5
0	1	0	0	0	29/60
0	0	1	0	0	-67/120
0	0	0	1	0	1/15
0	0	0	0	1	13/120

-4(R3) + R1 ---> R1

1	-2	0	0	0	61/30
0	1	0	0	0	29/60
0	0	1	0	0	-67/120
0	0	0	1	0	1/15
0	0	0	0	1	13/120

-2(R2) + R1 ---> R1

1	0	0	0	0	3
0	1	0	0	0	29/60
0	0	1	0	0	-67/120
0	0	0	1	0	1/15
0	0	0	0	1	13/120