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Point Quotient Group Incidence Geometry Fraction Polynomial Ring Valency Zero Of A Function

1	Incidence Geometry
1	Point
1	Polynomial Ring
1	Fraction
1	Quotient Group
2	Zero Of A Function
2	Valency

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Lagrange Interpolation Formula

Let $(x_1,y_1), (x_2,y_2),\dotsc,(x_n,y_n)$ be points in the plane ( $x_i\neq x_j$ for $i\neq j$ ). Then there exists a unique polynomial of degree at most such that for $i=1,\ldots,n$ .

Such polynomial can be found using Lagrange's interpolation formula:

$\displaystyle p(x)=\frac{f(x)}{(x-x_1)f'(x_1)}y_1+\frac{f(x)}{(x-x_2)f'(x_2)}y_2+\cdots+\frac{f(x)}{(x-x_n)f'(x_n)}y_n$

where $f(x)=(x-x_1)(x-x_2)\cdots(x-x_n)$ .

To see this, notice that the above formula is the same as

$\displaystyle p(x)$	$\displaystyle = y_1 \frac{(x-x_2)(x-x_3)\dots(x-x_n)}{(x_1-x_2)(x_1-x_3)\dots(x_1-x_n)} + y_2 \frac{(x-x_1)(x-x_3)\dots(x-x_n)}{(x_2-x_1)(x_2-x_3)\dots(x_2-x_n)}$
	$\displaystyle \phantom{=}\qquad+\dots+y_n \frac{(x-x_1)(x-x_2)\dots(x-x_{n-1})}{(x_n-x_1)(x_n-x_2)\dots(x_n-x_{n-1})}$

and that for all , every numerator except one vanishes, and this numerator will be identical to the denominator, making the overall quotient equal to 1. Therefore, each equals .