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# Polynomial Interpolation

Suppose you have a system of 50 linear equations. It's tedious and impractical to solve them by hand. Polynomial interpolation is your better alternative.

Consider the following system of linear equations: \[\begin{cases} 1&+&a&+&b&+&c&+&d&+&e&=&1,\\ 32&+&16a&+&8b&+&4c&+&2d&+&e&=&2,\\ 243&+&81a&+&27b&+&9c&+&3d&+&e&=&3,\\ 1024&+&256a&+&64b&+&16c&+&4d&+&e&=&4,\\ 3125&+&625a&+&125b&+&25c&+&5d&+&e&=&5.\\ \end{cases}\]

Evaluate \[ \left|\dfrac{8bcd}{125ae}\right|.\]

A polynomial \(f(x)\) has degree \(8\) and \(f(i)=2^i\) for \(i=0,1,2,3,4,5,6,7,8.\) Find \(f(9).\)

Let \( a,b,c \) be complex numbers satisfying

\( (a+1)(b+1)(c+1) = 1\)

\( (a+2)(b+2)(c+2) = 2\)

\( (a+3)(b+3)(c+3) = 3\)

Find \( (a+4)(b+4)(c+4) \).

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