Polynomial question. (1 Viewer)

[~shinigami~]

I've been going over some polynomial notes for my half-yearly and I came a cross something that I don't quite understand. I hope someone would kindly explain it to me.

I don't really understand the solution.

Thank you so much in advance.

 

[pLuvia]

x⁵+8x²=0
x²(x³+8)=0
Clearly x=0 is a solution

Basically their method of deriving the answers is using logic, since there are three answers, if you plot them on an argand diagram all three roots will be evenly spaced out by 2pi/3

Using an algebraic approach

You have x³=-8
Now using the method you use for complex roots of unity
x³=-8cis(0)
x=-2cis(2kpi/3), since it can be rotated 2kpi without changing the actual results, and where kER, 0,+1

When
k=0, x=-2
k=1, x=-2cis(2pi/3)
k=-1, x=-2cis(-2pi/3)

Which tidies up to become the solutions
0, -2, 1+isqrt(3)

 

[~shinigami~]

x⁵+8x²=0
x²(x³+8)=0
Clearly x=0 is a solution

Basically their method of deriving the answers is using logic, since there are three answers, if you plot them on an argand diagram all three roots will be evenly spaced out by 2pi/3

Using an algebraic approach

You have x³=-8
Now using the method you use for complex roots of unity
x³=-8cis(0)
x=-2cis(2kpi/3), since it can be rotated 2kpi without changing the actual results, and where kER, 0,+1

When
k=0, x=-2
k=1, x=-2cis(2pi/3)
k=-1, x=-2cis(-2pi/3)

Which tidies up to become the solutions
0, -2, 1+isqrt(3)

Thank you so much pLuvia, I get it now.

For some reason in my head, I kept thinking x³=-8cis(0) was x³=-8cis(pi).

Stupid me.