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# Find the equation of a polynomal with the following information

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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.

Aug 17, 2017

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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))

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