Polynomial Interpolation

Challenge Quizzes

Polynomial Interpolation: Level 5 Challenges


Consider the following system of linear equations:

\[\begin{cases} \begin{array}{rcrcrcrcrcrcrl} 1&+&a&+&b&+&c&+&d&+&e&=&1\\ 32&+&16a&+&8b&+&4c&+&2d&+&e&=&2\\ 243&+&81a&+&27b&+&9c&+&3d&+&e&=&3\\ 1024&+&256a&+&64b&+&16c&+&4d&+&e&=&4\\ 3125&+&625a&+&125b&+&25c&+&5d&+&e&=&5&.\\ \end{array} \end{cases}\]


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

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

Find the number of positive integers \(n\) such that \(2 \leq n \leq 100\) and there exists a polynomial \(f(x)\) with real coefficients and degree \( <n\) such that for all integers \(x,\) \(f(x)\) is an integer if and only if \(x\) is not a multiple of \(n.\)

Suppose \(f(x)\) is a degree-\(8\) polynomial such that \(f(2^i)=\frac{1}{2^i}\) for all integers \( 0 \leq i \leq 8\). If \(f(0)= \frac{a}{b}\), where \(a\) and \( b\) are coprime positive integers, what is the value of \(a+b?\)


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