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# Help! Polynomials

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Find all (real or nonreal)  z satisfying

$$(z - 3)^4 + (z - 5)^4 = -8$$

Jul 22, 2024

#1
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First, let's expand out everything. I'm not going to show the steps, but we eventually get

$$2z^4-32z^3+204z^2-608z+706=-8\\2z^4-32z^3+204z^2-608z+714=0$$

Now, since every number is divisible by 2, we get factor out 2 to get

$${2\left(z^{4}-16z^{3}+102z^{2}-304z+357\right)}=0$$

Now, note that $$z^{4}-16z^{3}+102z^{2}-304z+357=(z^{2}-8z+17)(z^{2}-8z+21)$$

This means that we have the equation

$$2\left(z^{2}-8z+17\right)\left(z^{2}-8z+21\right)=0$$

Since the equation is equal to 0, we get two equations. We have

$$z^{2}-8z+17=0,\qquad z^{2}-8z+21=0$$

Using the quadratic equation on both formulas, we find 4 values of z.

We have

$$z=4+i\\ z=4-i\\ z=4+\sqrt{5}\cdot (i)\\ z=4+\sqrt{5}\cdot (-i)$$

This was a very basic explenation of the problem.

I can go in dephth more if you want to.

Thanks! :)

Jul 22, 2024
edited by NotThatSmart  Jul 22, 2024
#2
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Thank you! I just really want the methods.

MeldHunter  Jul 23, 2024