Complex number questions: If P(z) = z^3 - z^2 - 4,....

[Guest]

If P(z) = z^3 - z^2 -4 how can I find the linear factors?
What is the remainder when P(z) is divided by z +i?

 

[Denis]

Re: Complex number questions

What is the remainder when z^3 - z^2 - 4 is divided by z + i?

I'll give it to you this way:
[z^2 - z(1+i) + i(1+i)] * (z + i) + (i - 3) = z^3 - z^2 - 4

If you don't "get it", then ask your teacher to teach you long division :shock:

 

[pka]

Here is yet another way to look at the question.

\(\displaystyle \L
P( - i) = - 3 + i .\)

 

[soroban]

Re: Complex number questions

Hello, americo74!

If \(\displaystyle P(z) \:= \:z^3\,-\,z^2\,-\,4\), how can I find the linear factors?

The Factor Theorem applies to complex polynoials, too.

Since \(\displaystyle z\,=\,2\) is a zero of the polynomial, \(\displaystyle \,(z\,-\,2)\) is a factor.

Then: \(\displaystyle \,P(x)\:=\z\,-\,2)(z^2\,+\,z\,+\,2)\)

\(\displaystyle \;\;\)and use the Quadratic Formula to find the linear factors.

What is the remainder when \(\displaystyle P(z)\) is divided by \(\displaystyle z+i\)?

The Remainder Theorem also applies here . . . as pka pointed out.