Complex Number Equations: Solving for Roots and Expressing in Standard Form

[Michael_Light]

Homework Statement

Under the section of complex number, i faced 2 questions which i couldn't answer... Here they go...

-Show that x=1-2i is a root of the equation x³-3x²+7x-5=0. Hence, find all the roots of the equation.


-Express http://img832.imageshack.us/img832/7916/msp180019e92f47ie4d5afg.gif in the form a+ib, where a>0.

Homework Equations

for both question, ''i'' represents imaginary number

The Attempt at a Solution

I tried both for hour but couldn't solve them...

 

[Char. Limit]

For the first one, you'll need polynomial long division. Do you know how to do that?

For the second one, I would first express your complex number in polar form.

 

[Michael_Light]

For the first one, you'll need polynomial long division. Do you know how to do that?

For the second one, I would first express your complex number in polar form.

Is it possible for you to explain briefly about polynomial long division and complex number in polar form... cause i have no ideas what are they... if can please show me step-by-step working... ><

 

[dextercioby]

Polynomial long division sounds nasty. I would use the fact that the 3rd order polynomial with real coefficients must necessarily have a real solution and the 2 complex ones are conjugate one to another.

As for the second point, i would have to find the A from

$$1+i\sqrt{3} = A^2 = (a+ib)^2$$

 

[tiny-tim]

Hi Michael!

-Show that x=1-2i is a root of the equation x³-3x²+7x-5=0. Hence, find all the roots of the equation.

"Show" means that you can assume that it's the answer …

so just put it into the LHS, and see whether that equals 0 (ie, what is (1 - 2i)³ etc?)

-Express http://img832.imageshack.us/img832/7916/msp180019e92f47ie4d5afg.gif in the form a+ib, where a>0.

(have a square-root: √ )

start by writing 1 + i√3 in the form re^iθ (that's polar form), ie r is the magnitude, and θ is the angle from the x-axis