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A cubic polynomial \(f\) satisfies \(f(0)=0, f(1)=1, f(2)=2, f(3)=4\). What is \(f(5)\)?

 Apr 16, 2019
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\(f(x)=a x^3 + b x^2 + c x + d\\ f(0)=0 \Rightarrow d=0\)

 

\(\text{see if you can figure out how I use the rest of the conditions to derive the following}\\ \begin{pmatrix}1 &1 &1\\8 &4 &2 \\ 27 &9 &3\end{pmatrix}\begin{pmatrix}a\\b\\c\end{pmatrix}=\begin{pmatrix}1\\2\\4\end{pmatrix}\)

 

Solving this system using Gaussian eliminations obtains

 

\((a,b,c) =\Large \left(\frac{1}{6},-\frac{1}{2},\frac{4}{3}\right)\\ f(5) = (a.b.c)\cdot (125,25,5) = 15\)

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 Apr 16, 2019

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