quicker way to determine if polynomial is unfactorable?

[mikexz]

Hi,

The problem is I often have to factor very large high degree polynomials (to the power of 5 or 6 or 7...) in a very short period of time (tests, exams) I often run out of time before I can find the factors or find out it is unfactorable. I usually start with the remainder and factor theorem, guessing numbers to plug into x but there are so many (sometimes 10 or 23 different numbers that MIGHT make f(x)=o). The most frustrating part is going through all those numbers to find out that it is unfactorable. Is there a faster way to factoring large polynomials and to find out if it is unfactorable?

thanks

 

[stapel]

Other than graphing and looking for x-intercepts (which will, of course, only tell you about linear real factors), I can't think of any particular "time saving" methods. Sorry!

Eliz.

 

[masters]

You could use Descartes' rule of signs to check for the number of possible positive, negative, and complex roots.

You could use the rational roots theorem to test values and find rational zeros.

Dale

 

[Deleted member 4993]

Hi,

The problem is I often have to factor very large high degree polynomials (to the power of 5 or 6 or 7...) in a very short period of time (tests, exams) I often run out of time before I can find the factors or find out it is unfactorable. I usually start with the remainder and factor theorem, guessing numbers to plug into x but there are so many (sometimes 10 or 23 different numbers that MIGHT make f(x)=o). The most frustrating part is going through all those numbers to find out that it is unfactorable. Is there a faster way to factoring large polynomials and to find out if it is unfactorable?

thanks

Unlike quadratic, cubic, and quartic polynomials, the general quintic cannot be solved algebraically in terms of a finite number of additions, subtractions, multiplications, divisions, and root extractions, as rigorously demonstrated by Abel (Abel's impossibility theorem) and Galois. However, certain classes of quintic equations can be solved in this manner.

That means - in general you cannot factorize fifth order equation ( let alone "to the power of .. 6 or 7...") - algebraically. At this point, we generally resort to numerical methods (where the answers are approximate).