A polynomial function of degree $n$ is a function of the form
$$ P(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + \cdots + a_{1}x + a_{0}$$
where $n$ is a positive integer and $a_{n} \neq 0$.
Every polynomial with real coefficients can be factored over the real numbers into linear factors or irreducible quadratic factors. This major result is due to the division, remainder and factor theorems.
The Factor Theorem
The factor theorem states:
$$ P(c) = 0 \textrm{ if and only if } (x-c) \textrm{ is a factor of the polynomial } P(x). $$
Factoring Using Synthetic or Long Division
To factor a polynomial, we can often use the procedure below.
Substitute $x$-values into $P(x)$ until you obtain $P(c)=0$ for some $x=c$. Good $x$-values to choose are positive and negative factors of the constant term $a_{0}$.
By the Factor Theorem $(x-c)$ is a factor of $P(x)$.
Use synthetic division or polynomial long division to calculate $P(x) \div (x-c)=Q(x)$.
Then $P(x) = (x-c)\cdot Q(x)$.
Factor $Q(x)$, if possible. If you are unable to factor $Q(x)$ by inspection, you may repeat the above procedure on $Q(x)$.
Note: There may be instances where it is very difficult to find $c$ satisfying $P(c)=0$. In these cases, the above method may not be suitable for factoring your polynomial and another method may need to be used.
Before proceeding, you may wish to review long division and synthetic division from
Section 1.3.
Factoring polynomials.