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**+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

#2**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