Can EVERY polynomial function be factored with long division?

[abel muroi]

IS there any polynomial function that can't be factored with long division?

 

[stapel]

Any prime polynomial, by definition, cannot be factored by any method.

 

[Ishuda]

IS there any polynomial function that can't be factored with long division?

It depends on how you define the factors. If, for example, you require all coefficients to be integers (or, more generally, rational numbers), then as staple mentioned any prime polynomial, by definition, cannot be factored by any method. The understood 'over the integers/rationals' is understood as part of the definition of prime polynomial.

For example, although we know the factors of x²-2 are (x-\(\displaystyle \sqrt{2}\)) and (x+\(\displaystyle \sqrt{2}\)), the \(\displaystyle \sqrt{2}\) is not a rational number and x²-2 is a prime polynomial (over the rationals). However you can factor it by long division.

 

[abel muroi]

Abel, suggest you go here and read up:
http://www.mathsisfun.com/algebra/polynomials.html

Then perhaps ask your question in a clearer fashion...

Ok i'll ask the question in another way.

If i am asked to graph a polynomial function (that isn't factored), should I always use the long division method to factor the polynomial function to get the roots of the graph?

 

[Steven G]

Ok i'll ask the question in another way.

If i am asked to graph a polynomial function (that isn't factored), should I always use the long division method to factor the polynomial function to get the roots of the graph?

Abel, I am not sure if you are asking a clear question. For example if I ask you to divide 36 can you do that? What does it mean? You should ask, divide 36 by what? Just like numbers, you divide polynomialS.
If you do not know the roots than what are you dividing by to get the roots? Are you using the rational roots theorem? Tell us exactly what you are doing/want

 

[stapel]

If i am asked to graph a polynomial function (that isn't factored), should I always use the long division method to factor the polynomial function to get the roots of the graph?

No. You could use synthetic division. You could, if the polynomial is amenable, use "by inspection" methods. The tools you use will vary with the information provided (and with your own inclinations).