• Jul 10th 2009, 11:14 AM
flower3
use complex analysis to factor the following polynomials into a product of two quadratic polynomials:
$x{^4} + 1$
---------------------------------------------------------------------------------------------------------------
sketch :
$\mid Arg z \mid \leq \frac{\pi}{4}$ and determine if the set is :
open,closed,bounded,connected ,simply connected , domain,compact or region
the given set is :
closed,connected ,simply connected ,and region
but not :
open,bounded,domain,and compact
my question : if the question : arg z not Arg z"principle argument" the answer is the same or not ???
• Jul 10th 2009, 02:55 PM
AlephZero
From the law $a^2-b^2=(a+b)(a-b),$ the fact that $i^2=-1,$ and the fact that $x^4+1=x^4-(-1),$ you should be able to prove the first part easily.
• Jul 11th 2009, 08:05 PM
Soroban
Hello, flower3!

Quote:

Use complex analysis to factor the following polynomial
into a product of two quadratic polynomials: . $x{^4} + 1$

Complex analysis? . . . It requires only Algebra 1.

Add and subtract $2x^2\!:\quad x^4 \:{\color{red}+ \:2x^2} + 1 \:{\color{red}- \:2x^2} \;=\;\left(x^2+1\right)^2 - \left(\sqrt{2}\:\!x\right)^2$

We have a difference of squares:

. . $\left[(x^2+1) - \sqrt{2}\:\!x\right]\,\left[(x^2+1) + \sqrt{2}\:\!x\right]$

. . . $=\;\left(x^2 - \sqrt{2}\:\!x + 1\right)\,\left(x^2 + \sqrt{2}\:\!x + 1\right)$

• Jul 11th 2009, 09:49 PM
malaygoel
Quote:

Originally Posted by flower3
use complex analysis to factor the following polynomials into a product of two quadratic polynomials:
$x{^4} + 1$

$x^4 + 1$

= $x^4-i^2$

= $(x^2+i)(x^2-i)$
• Jul 12th 2009, 04:59 AM
mr fantastic
Quote:

Originally Posted by malaygoel
$x^4 + 1$

= $x^4-i^2$

= $(x^2+i)(x^2-i)$

I suspect the OP meant two quadratic polynomials with real coefficients.

Quote:

Originally Posted by Soroban
Hello, flower3!

Complex analysis? . . . It requires only Algebra 1.

Add and subtract $2x^2\!:\quad x^4 \:{\color{red}+ \:2x^2} + 1 \:{\color{red}- \:2x^2} \;=\;\left(x^2+1\right)^2 - \left(\sqrt{2}\:\!x\right)^2$

We have a difference of squares:

. . $\left[(x^2+1) - \sqrt{2}\:\!x\right]\,\left[(x^2+1) + \sqrt{2}\:\!x\right]$

. . . $=\;\left(x^2 - \sqrt{2}\:\!x + 1\right)\,\left(x^2 + \sqrt{2}\:\!x + 1\right)$

Unfortunately, in an exam setting, if the question gives the instruction to use complex analysis then a correct answer using an alternative technique will not score any marks. Not even an answer mark.

Quote:

Originally Posted by flower3
use complex analysis to factor the following polynomials into a product of two quadratic polynomials:
$x{^4} + 1$
[snip]

Start by finding the four complex roots of -1. Can you do this? (If not, see http://www.mathhelpforum.com/math-he...ex-number.html, for example).
• Jul 12th 2009, 05:19 AM
mr fantastic
Quote:

Originally Posted by flower3
[snip]
sketch :
$\mid Arg z \mid \leq \frac{\pi}{4}$ and determine if the set is :
open,closed,bounded,connected ,simply connected , domain,compact or region
the given set is :
closed,connected ,simply connected ,and region
but not :
open,bounded,domain,and compact
my question : if the question : arg z not Arg z"principle argument" the answer is the same or not ???

Your question can be easily answered by noting that $\mid \text{Arg} (z) \mid \leq \frac{\pi}{4}$ is equivalent to $- \frac{\pi}{4} \leq \text{Arg} (z) \leq \frac{\pi}{4}$ ....
• Jul 12th 2009, 11:56 AM
flower3
Quote:

Originally Posted by mr fantastic
Your question can be easily answered by noting that $\mid \text{Arg} (z) \mid \leq \frac{\pi}{4}$ is equivalent to $- \frac{\pi}{4} \leq \text{Arg} (z) \leq \frac{\pi}{4}$ ....

(Itwasntme) i know that , but i mean if the question $\mid arg z \mid \leq \frac{\pi}{4}$ is the answer the same or not ???
closed,connected ,simply connected ,and region
but not :
open,bounded,domain,and compact (Wait)
• Jul 12th 2009, 12:00 PM
flower3
Quote:

Originally Posted by mr fantastic
I suspect the OP meant two quadratic polynomials with real coefficients.

Unfortunately, in an exam setting, if the question gives the instruction to use complex analysis then a correct answer using an alternative technique will not score any marks. Not even an answer mark.

Start by finding the four complex roots of -1. Can you do this? (If not, see http://www.mathhelpforum.com/math-he...ex-number.html, for example).

Thanks !
if i find one of theses roots and then multiply it by its conjugate then by using long division i find the 2nd quadratic polynomial "true or not ???"
• Jul 12th 2009, 06:23 PM
mr fantastic
Quote:

Originally Posted by flower3
Thanks !
if i find one of theses roots and then multiply it by its conjugate then by using long division i find the 2nd quadratic polynomial "true or not ???"

The roots will have the form $z = \alpha_1$, $z = \overline{\alpha_1}$, $z = \alpha_2$ and $z = \overline{\alpha_2}$.

Therefore the two quadratic factors you require will obviously be $(z - \alpha_1)(z - \overline{\alpha_1})$ and $(z - \alpha_2)(z - \overline{\alpha_2})$.

Quote:

Originally Posted by flower3
(Itwasntme) i know that , but i mean if the question $\mid arg z \mid \leq \frac{\pi}{4}$ is the answer the same or not ???
closed,connected ,simply connected ,and region
but not :
open,bounded,domain,and compact (Wait)

The answer is contained in post #6.