Factoring a 3rd order polynomial

[rowardHoark]

Factoring a 4th order polynomial

Homework Statement

Example:
[itex](jw)^{3}+6(jw)^{2}+5jw+30=0[/itex] can be re-written into [itex]6(5-w^{2})+jw(5-w^{2})[/itex]. The fact that there are two identical [itex](5-w^{2})[/itex] is a desirable outcome. Imaginary number [itex]j=\sqrt{-1}[/itex] becomes -1 when raised to the power of 2.

Homework Equations

The problem is to transform [itex] (jw)^{4}+7(jw)^{3}+59(jw)^2+98(jw)+630=0 [/itex] in a similar manner.

The Attempt at a Solution

So far I have been unsuccessful.

[itex]w^{4}-7jw^{3}-59w^{2}+98jw+630=0[/itex]

[itex](w^{4}-59w^{2})+7(-jw^{3}+14jw+90)=0[/itex]

 

[CEL]

Homework Statement

Example:
[itex](jw)^{3}+6(jw)^{2}+5jw+30=0[/itex] can be re-written into [itex]6(5-w^{2})+jw(5-w^{2})[/itex]. The fact that there are two identical [itex](5-w^{2})[/itex] is a disirable outcome. Imaginary number [itex]j=\sqrt{-1}[/itex] becomes -1 when raised to the power of 2.

Homework Equations

The problem is to transform [itex] (jw)^{4}+7(jw)^{3}+59(jw)^2+98(jw)+630=0 [/itex] in a similar manner.

The Attempt at a Solution

So far I have been unsuccessful.

[itex]w^{4}-7jw^{3}-59w^{2}+98jw+630=0[/itex]

[itex](w^{4}-59w^{2})+7(-jw^{3}+14jw+90)=0[/itex]

Try

[itex](w^4 - 59 w^2 + 630) + jw(14 - w^2)[/itex]

Then divide [itex](w^4 - 59 w^2 + 630)[/itex] by [itex](14 - w^2)[/itex]

 

[rowardHoark]

Try

[itex](w^4 - 59 w^2 + 630) + jw(14 - w^2)[/itex]

Then divide [itex](w^4 - 59 w^2 + 630)[/itex] by [itex](14 - w^2)[/itex]

Thank you, CEL.

The answer is [itex]-(14-w^{2})(w^{2}-45)+7jw(14-w^{2})=0[/itex]

If designing a controller using Ziegler-Nichols second method, would I pick [itex]\omega=\sqrt{14}[/itex] or [itex]\omega=\sqrt{45}[/itex] as my value to calculare [itex]P_{cr}=\frac{2\Pi}{\omega} [/itex]?

 

[The Electrician]

Try

[itex](w^4 - 59 w^2 + 630) + jw(14 - w^2)[/itex]

Then divide [itex](w^4 - 59 w^2 + 630)[/itex] by [itex](14 - w^2)[/itex]

There's one tiny error here:

[itex](w^4 - 59 w^2 + 630) + jw(14 - w^2)[/itex]

should be:

[itex](w^4 - 59 w^2 + 630) + 7jw(14 - w^2)[/itex]

A good systematic method for problems like this is shown in the attachment.

 

[DeeNos]

(w^{2}-59)(w^{2}+14j)+7j(w^{2}+14j)=0

At this point, I am unable to simplify the expression any further. It is important to note that factoring a 3rd order polynomial and a 4th order polynomial are two different processes. In fact, factoring a 4th order polynomial is more complex and often involves using techniques such as grouping, the rational root theorem, and the quadratic formula. In this case, the polynomial is not easily factorable and may require the use of a computer program or graphing calculator to find the roots. It is also possible that the polynomial may not have any real roots and will need to be factored using complex numbers. As a scientist, it is important to carefully analyze and approach each problem with the appropriate techniques and tools.