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Find the equation of a polynomial f(x) of degree 5 with zeros: x=200, x=100, x=0, x=-100, x=-200 and f(5)=5.  Show what the maxiums and minimums are. Please show each step and graph with all zero points and maxium and minimum points labeled.

gibsonj338  Aug 17, 2017
 #1
avatar+178 
+1

Given that \(f(5)=5\) and it crosses \(y=0\) at the following points:

\(x=200, x=100,x=0,x=-100,x=-200\)

Since the crossing points of this function is rational, I deduct that the function is factorizable.

Perform reverse-factorization:

The function need to be in this form:

\(g(x)=(x+200)(x+100)(x+0)(x-100)(x-200)\)

For it to have roots at \(x=200, x=100, x=0, x=-100,x=-200\)

There are a total of five zero-points at:

\(1.x=-200 , y=0\)

\(2.x=-100 , y=0\)

\(3.x=0 , y=0\space(Origin)\)

\(4.x=100 , y=0\)

\(5.x=200, y=0\)

There are a total of four critical points at:

Local Maxima:

\(1.x=\sqrt{15000-1000\sqrt{145}} , y=200000000\left(\sqrt{5}+5\sqrt{29}\right)\sqrt{30-2\sqrt{145}}\)

\(2.x=-\sqrt{15000+1000\sqrt{145}} , y=-200000000\left(\sqrt{5}-5\sqrt{29}\right)\sqrt{30+2\sqrt{145}}\)

Local Minima:

\(3.x=\sqrt{15000+1000\sqrt{145}} , y=\left(\sqrt{5}-5\sqrt{29}\right)\sqrt{30+2\sqrt{145}}\)

\(4.x=-\sqrt{15000-1000\sqrt{145}} , y=-200000000\left(\sqrt{5}+5\sqrt{29}\right)\sqrt{30-2\sqrt{145}}\)

Since \(f(5)=5\), Just divide every y-value of maximas and minimas above by a factor of \(g(5)/5=398750625\)

\(f(x)=\frac{1}{398750625}\left(x+200\right)\left(x+100\right)x\left(x-100\right)\left(x-200\right)\)

Q.E.D.

(I bet you just randomly typed the numbers in, didn't you? (Because the \(x\) and \(y\) values are pretty ugly to be honest))

Jeffes02  Aug 17, 2017
edited by Jeffes02  Aug 17, 2017
edited by Jeffes02  Aug 17, 2017
 #2
avatar+1841 
0

I did just type in the numbers. :D.  By the way, not to sound criticizing, you did not graph the answer as per the question.

gibsonj338  Aug 17, 2017
 #3
avatar+178 
+2

(After downloading printscreen and photo-editing plugins later...)
(Wait I think I messed the pictures up...)

Jeffes02  Aug 17, 2017
edited by Jeffes02  Aug 17, 2017

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