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Let $f$ be a cubic polynomial such that $f(0) = 5$, $f(-1) = -8$, $f(2) = 18$, and $f(5)=21$.  What is the sum of the coefficients of $f$?

 Feb 17, 2024
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let \(f(x)=ax^3+bx^2+cx+d\) using the information we can get a system of equations to solve for a,b,c and d then add them.

\(f(0)=0\cdot a+0\cdot b+0\cdot c+d=5\\ \Rightarrow d=5\)

\(f(-1)=(-1)^3a+(-1)^2b+(-1)c+d=-8\\ \Rightarrow -a+b-c=-13\)

\(f(2)=2^3a+2^2b+2c+d=18\\ \Rightarrow 8a+4b+2c=13\)

\(f(5)=5^3a+5^2b+5c+d=21\\ \Rightarrow 125a+25b+5c=16\)

Solving the system of equations(i just kept them in a random online site and double checked them) we get:
\(a=\frac{8}{45}\\ b=-\frac{211}{90}\\ c=\frac{943}{90}\\ d=5\\ \LARGE\boxed{a+b+c+d=\frac{599}{45}=13\frac{14}{45}}\)

 Feb 17, 2024

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