# How can you find the factors of a polynomial?

Mar 1, 2017

This question is not easy to answer as a generic case

#### Explanation:

There is a vast number of variations in structure of polynomials. So in a sense it is bit like asking:

How long is a piece of string?

Basically you look for common individual variable, combinations of variables and or constants that can be 'extracted' from the originally presented polynomial. They have to be such that when they are all multiplied together you end up back at the original polynomial.

Mar 1, 2017

Some thoughts...

#### Explanation:

Polynomials behave a bit like numbers in some ways. You can add, subtract and multiply them. You can attempt to divide one polynomial by another, giving a quotient and remainder. The remainder will be zero if the dividend is an exact multiple of the divisor.

Here's an example long division of ${x}^{3} + {x}^{2} - x - 1$ by $x - 1$ ...

In theory, any polynomial in one variable with real coefficients can be factorised into a product of linear and quadratic factors with real coefficients. In practice this can be very hard or even inexpressible in terms of square and cube roots, etc.

For example, ${x}^{6} + {x}^{3} + 1$ has factors which are expressible in terms of cube roots of complex numbers, or in terms of trigonometric functions, but not in terms of real square and cube roots.

Even worse is something like ${x}^{5} + 4 x + 2$ whose zeros (and therefore the coefficients of its factors) are not expressible in terms of $n$th roots or elementary functions.

You can use things like the rational zeros theorem to find rational zeros and factors, but it does not help much with irrational zeros and associated factors.