Question 1176849
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We are used to looking at ordered pairs as *[tex \Large \(x,y\)], but for this problem, it would be helpful to consider an ordered pair to represent *[tex \Large \(x,\f(x)\)] which is to say that the first value is a value in the domain of *[tex \Large f] and the second value is the value of the function at that input value.  In other words, if *[tex \Large (1,6)] is a point on the graph of some function *[tex \Large f], then *[tex \Large f(1)\ =\ 6].


Given that, if


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f(x)\ =\ ax^4\,+\,bx^3\,+\,cx^2\,+\,dx\,+\,e]


and the point *[tex \Large \(1,6\)] is on the graph of the function then it must be true that:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 6\ =\ a(1)^4\,+\,b(1)^3\,+\,c(1)^2\,+\,d(1)\,+\,e]


Which simplifies to


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ a\ +\ b\ +\ c\ +\ d\ +\ e\ =\ 6]


Similarly,


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 3\ =\ a(2)^4\,+\,b(2)^3\,+\,c(2)^2\,+\,d(2)\,+\,e]


Which simplifies to


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 16a\ +\ 8b\ +\ 4c\ +\ 2d\ +\ e\ =\ 3]


And so on for the other 3 given points.


Once you have the 5 linear equations in the 5 variables a, b, c, d, and e, solve the 5X5 system for the 5 values and use these values for the coefficients when writing the specific definition of *[tex \Large f(x)].  There are several on-line linear system solvers and I would recommend using one of them since the coefficients in your linear system are going to get very big before you are through and the arithmetic will be uglier than a mud fence.  Good luck.

																
John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it
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