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Algebra Level 3

$z^4=5(z-1)(z^2-z+1)$

If $z$ is a complex number satisfying above equation, find the number of solutions for $z$ to the above equation.

2 4 3 1

1 solution

Eddy Li
Dec 22, 2015

Plz... Don't fall on the trap and create a graph, because it will produce only 2 REAL solutions. A good idea is to understand the nature of polynomials.

The polynomial should have been $z^{ 4 } = 5 (z -1)(z^{ 2 }- z+1)$ $z^{ 4 }= 5(z^{ 3 }- 2z^{ 2 }-1)$ $z^{ 4 } - 5z^{ 3 }+10z^{ 2 } - 1=0$

As you can see, the highest power of this polynomial is 4 so there should be 4 solutions to this equation.

Extension Problem I don't know the exact solutions to this polynomial so I let you guys find out!

"The highest power of this polynomial is 4 so there should be 4 solutions to this equation." This is incorrect reasoning.The correct statement is "The highest power of this polynomial is 4 so there should be ' at most ' 4 solutions to this equation."

Nihar Mahajan - 5 years, 5 months ago

Yeah, I forgot about that, but then can you just prove there are no double roots in this solution though?

Eddy Li - 5 years, 5 months ago

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