Find the polynomial of degree 4 whose graph goes through the points (-3,-218),(-1,4),(0,10),(2,22) and (3,-32).f(x) = ___x^4 + ____x^3 + _____x^2 +_____x + _______

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tiburtius | High School Teacher | (Level 2) Educator

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If you want to find function `f` whose graph goes through  a point `(x,y)`. That means you need to find solution of equation `f(x)=y`. If you have several points that means you need to find solution of system of equations.

In your case you want to find polynomial  `f(x)=ax^4+bx^3+cx^2+dx+e` and for that you need to solve the following system:

`(-3)^4a+(-3)^3b+(-3)^2c+(-3)d+e=-218`

`a-b+c-d+e=4`

`e=10`

`16a+8b+4c+2d+e=22`

`81a+27b+9c+3d+e=-32`

Now you have system of 5 linear equations with 5 variables which you can solve by using Gauss elimination or some other method. Solution to this system of equations is

`a=-2`, `b=3`, `c=3`, `d=4`, `e=10`

Hence, your polynomial is:

`f(x)=-2x^4+3x^3+3x^2+4x+10`

 

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