We'll find the interpolating polynomial passing through the points ( 1 , − 6 ) {\displaystyle (1,-6)} , ( 2 , 2 ) {\displaystyle (2,2)} , ( 4 , 12 ) {\displaystyle (4,12)} , using the Lagrange method .
We first use the formula to write the following:
p ( x ) = − 6 ( x − 2 ) ( 1 − 2 ) ( x − 4 ) ( 1 − 4 ) + 2 ( x − 1 ) ( 2 − 1 ) ( x − 4 ) ( 2 − 4 ) + 12 ( x − 1 ) ( 4 − 1 ) ( x − 2 ) ( 4 − 2 ) {\displaystyle p(x)=-6{\frac {(x-2)}{(1-2)}}{\frac {(x-4)}{(1-4)}}+2{\frac {(x-1)}{(2-1)}}{\frac {(x-4)}{(2-4)}}+12{\frac {(x-1)}{(4-1)}}{\frac {(x-2)}{(4-2)}}}
After some simplification, we get:
p ( x ) = − 2 ( x − 2 ) ( x − 4 ) − 1 ( x − 1 ) ( x − 4 ) + 2 ( x − 1 ) ( x − 2 ) {\displaystyle p(x)=-2(x-2)(x-4)-1(x-1)(x-4)+2(x-1)(x-2)}
p ( x ) = − 2 ( x 2 − 6 x + 8 ) − 1 ( x 2 − 5 x + 4 ) + 2 ( x 2 − 3 x + 2 ) {\displaystyle p(x)=-2(x^{2}-6x+8)-1(x^{2}-5x+4)+2(x^{2}-3x+2)}
And our answer:
p ( x ) = − x 2 + 11 x − 16 {\displaystyle p(x)=-x^{2}+11x-16} .
Adding a point
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Now we'll add a point to our data set, and find a new interpolating polynomial. Let us add the point ( 3 , − 10 ) {\displaystyle (3,-10)} to our set. Starting over with the Lagrange formula, we write:
p ( x ) = − 6 ( x − 2 ) ( 1 − 2 ) ( x − 4 ) ( 1 − 4 ) ( x − 3 ) ( 1 − 3 ) + 2 ( x − 1 ) ( 2 − 1 ) ( x − 4 ) ( 2 − 4 ) ( x − 3 ) ( 2 − 3 ) + 12 ( x − 1 ) ( 4 − 1 ) ( x − 2 ) ( 4 − 2 ) ( x − 3 ) ( 4 − 3 ) − 10 ( x − 1 ) ( 3 − 1 ) ( x − 2 ) ( 3 − 2 ) ( x − 4 ) ( 3 − 4 ) {\displaystyle p(x)=-6{\frac {(x-2)}{(1-2)}}{\frac {(x-4)}{(1-4)}}{\frac {(x-3)}{(1-3)}}+2{\frac {(x-1)}{(2-1)}}{\frac {(x-4)}{(2-4)}}{\frac {(x-3)}{(2-3)}}+12{\frac {(x-1)}{(4-1)}}{\frac {(x-2)}{(4-2)}}{\frac {(x-3)}{(4-3)}}-10{\frac {(x-1)}{(3-1)}}{\frac {(x-2)}{(3-2)}}{\frac {(x-4)}{(3-4)}}}
Simplifying, we get:
p ( x ) = ( x − 2 ) ( x − 4 ) ( x − 3 ) + ( x − 1 ) ( x − 4 ) ( x − 3 ) + 2 ( x − 1 ) ( x − 2 ) ( x − 3 ) + 5 ( x − 1 ) ( x − 2 ) ( x − 4 ) {\displaystyle p(x)=(x-2)(x-4)(x-3)+(x-1)(x-4)(x-3)+2(x-1)(x-2)(x-3)+5(x-1)(x-2)(x-4)}
p ( x ) = ( x 3 − 9 x 2 + 26 x − 24 ) + ( x 3 − 8 x 2 + 19 x − 12 ) + 2 ( x 3 − 6 x 2 + 11 x − 6 ) + 5 ( x 3 − 7 x 2 + 14 x − 8 ) {\displaystyle p(x)=(x^{3}-9x^{2}+26x-24)+(x^{3}-8x^{2}+19x-12)+2(x^{3}-6x^{2}+11x-6)+5(x^{3}-7x^{2}+14x-8)}
And our polynomial is:
p ( x ) = 9 x 3 − 64 x 2 + 137 x − 88 {\displaystyle p(x)=9x^{3}-64x^{2}+137x-88} .