If $f(x)$ is the polynomial (coefficient of leading term is unity) in 'x' of least degree such that $f(1)=5 , f(2)=4, f(3)=3, f(4)=2, f(5)=1$
Then $f(0)= ?$
2026-04-02 06:20:09.1775110809
if f(x) is the polynomial (coeff of leadin term is unity) in 'x' of least degree such that f(1)=5 , f(2)=4, f(3)=3, f(4)=2, f(5)=1, then f(0)=?
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The naive approach via Lagrange yields $$f(0)=5\frac{(0-2)(0-3)(0-4)}{(1-2)(1-3)(1-4)}+4\frac{(0-1)(0-3)(0-4)}{(2-1)(2-3)(2-4)}+3\frac{(0-1)(0-2)(0-4)}{(3-1)(3-2)(3-4)}+2\frac{(0-1)(0-2)(0-3)}{(4-1)(4-2)(4-3)},$$ but this polynomial is just $-x+6$ ... Notice that the four points lay on a straight line hence the polynomial must have degree $4$ or more. Agent Vandermonde then finds $f(x)=x^4-10x^3+35x^2-51x+30$.