Factoring a polynomial means rewriting it as a product of simpler polynomial factors.
A complete factorization uses all available patterns until no further rational or integer factorization is possible.
1. Start with standard form
Before factoring, write the polynomial in descending powers:
\[
\begin{aligned}
x^3-6x^2+11x-6.
\end{aligned}
\]
Like terms should be combined before factoring begins.
2. Greatest common factor
The first step is usually to check whether every term has a common factor.
\[
\begin{aligned}
6x^4-12x^3+6x^2
&=
6x^2(x^2-2x+1).
\end{aligned}
\]
Then the remaining factor may be factored further:
\[
\begin{aligned}
x^2-2x+1
&=
(x-1)^2.
\end{aligned}
\]
3. Linear factors and roots
If \(r\) is a root of a polynomial, then \(x-r\) is a factor:
\[
\begin{aligned}
f(r)=0
\quad\Longrightarrow\quad
(x-r)\text{ is a factor of }f(x).
\end{aligned}
\]
For example, if \(f(1)=0\), then \((x-1)\) is a factor.
4. Rational root theorem
For a polynomial with integer coefficients:
\[
\begin{aligned}
a_nx^n+\cdots+a_1x+a_0,
\end{aligned}
\]
any rational root \(p/q\) in lowest terms must satisfy:
\[
\begin{aligned}
p&\mid a_0,\\
q&\mid a_n.
\end{aligned}
\]
This gives a finite list of possible rational roots to test.
5. Worked example: \(x^3-6x^2+11x-6\)
Possible integer roots are factors of \(6\):
\[
\begin{aligned}
\pm1,\ \pm2,\ \pm3,\ \pm6.
\end{aligned}
\]
Test \(x=1\):
\[
\begin{aligned}
f(1)
&=
1^3-6(1)^2+11(1)-6\\
&=
1-6+11-6\\
&=
0.
\end{aligned}
\]
Therefore, \((x-1)\) is a factor:
\[
\begin{aligned}
x^3-6x^2+11x-6
&=
(x-1)(x^2-5x+6).
\end{aligned}
\]
Factor the quadratic:
\[
\begin{aligned}
x^2-5x+6
&=
(x-2)(x-3).
\end{aligned}
\]
So:
\[
\begin{aligned}
\boxed{
x^3-6x^2+11x-6
=
(x-1)(x-2)(x-3)
}.
\end{aligned}
\]
6. Factoring quadratics
A quadratic has the form:
\[
\begin{aligned}
ax^2+bx+c.
\end{aligned}
\]
If it factors rationally, it can be written as:
\[
\begin{aligned}
ax^2+bx+c
&=
(px+q)(rx+s).
\end{aligned}
\]
The discriminant helps determine whether rational linear factors exist:
\[
\begin{aligned}
D=b^2-4ac.
\end{aligned}
\]
If \(D\) is a perfect square, the quadratic has rational roots and can factor into rational linear factors.
If \(D\) is not a perfect square, the quadratic is irreducible over the rationals.
7. Difference of squares
The difference of squares pattern is:
\[
\begin{aligned}
a^2-b^2
&=
(a-b)(a+b).
\end{aligned}
\]
Example:
\[
\begin{aligned}
x^2-49
&=
x^2-7^2\\
&=
(x-7)(x+7).
\end{aligned}
\]
8. Sum and difference of cubes
The cube patterns are:
\[
\begin{aligned}
a^3-b^3
&=
(a-b)(a^2+ab+b^2),\\
a^3+b^3
&=
(a+b)(a^2-ab+b^2).
\end{aligned}
\]
Examples:
\[
\begin{aligned}
x^3-27
&=
x^3-3^3\\
&=
(x-3)(x^2+3x+9),
\end{aligned}
\]
and
\[
\begin{aligned}
x^3+8
&=
x^3+2^3\\
&=
(x+2)(x^2-2x+4).
\end{aligned}
\]
9. Factoring by grouping
Grouping is useful when terms can be arranged into pairs with a common binomial factor.
\[
\begin{aligned}
3x^3+6x^2+2x+4
&=
3x^2(x+2)+2(x+2)\\
&=
(x+2)(3x^2+2).
\end{aligned}
\]
The common binomial factor is \((x+2)\).
10. Irreducible factors
A factor is irreducible over the rationals if it cannot be factored further using rational coefficients.
For example:
\[
\begin{aligned}
x^2+x+1
\end{aligned}
\]
has discriminant:
\[
\begin{aligned}
D&=1^2-4(1)(1)\\
&=-3.
\end{aligned}
\]
Since the discriminant is negative, it has no real roots and is irreducible over the rationals.
11. Checking a factorization
To check a factorization, expand the product. For example:
\[
\begin{aligned}
(x-1)(x-2)(x-3)
&=
(x^2-3x+2)(x-3)\\
&=
x^3-6x^2+11x-6.
\end{aligned}
\]
The expanded form matches the original polynomial, so the factorization is correct.
12. Formula summary
The table below uses plain text formulas in table cells to avoid raw LaTeX rendering problems.
13. Common mistakes
- Forgetting to factor out the GCF before using another method.
- Testing only positive roots and forgetting negative root candidates.
- Using the difference of squares pattern on a sum such as \(x^2+49\).
- Mixing up the signs in the sum and difference of cubes formulas.
- Leaving a quadratic unfactored without checking its discriminant.
- Not checking the final product by expanding it.
Key idea: factor systematically—standard form, GCF, special patterns, rational roots, quadratic checks, then irreducible factors.
