Algebra
# Polynomial Interpolation

Consider the following system of linear equations:

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

Evaluate

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

$f(x)$ has degree $8$ and $f(i)=2^i$ for $i=0,1,2,3,4,5,6,7,8.$

A polynomialFind $f(9).$

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