Simplifying and factoring are two connected algebra skills. Simplifying rewrites an expression in a cleaner form,
while factoring rewrites an expression as a product of smaller expressions.
1. Terms and like terms
A polynomial is made of terms. For example:
\[
\begin{aligned}
2x^2+8x-24
\end{aligned}
\]
has three terms: \(2x^2\), \(8x\), and \(-24\).
Like terms have the same variable part and the same exponent.
Example:
\[
\begin{aligned}
3x^2+5x-2x^2+7x
&=
(3x^2-2x^2)+(5x+7x)\\
&=
x^2+12x.
\end{aligned}
\]
2. Expanding before simplifying
Sometimes expressions contain parentheses. To simplify, expand first and then combine like terms.
\[
\begin{aligned}
(x+2)(x-3)
&=
x^2-3x+2x-6\\
&=
x^2-x-6.
\end{aligned}
\]
This calculator expands simple products of polynomial factors so the resulting polynomial can be simplified.
3. Greatest common factor
The first factoring step is often to look for a greatest common factor, or GCF.
The GCF is the largest factor shared by all terms.
Example:
\[
\begin{aligned}
6x^3-12x^2+18x
&=
6x(x^2-2x+3).
\end{aligned}
\]
Here, every term has a factor of \(6x\).
4. Factoring quadratics
A quadratic has the form:
\[
\begin{aligned}
ax^2+bx+c.
\end{aligned}
\]
When possible, it can be factored into two binomials:
\[
\begin{aligned}
ax^2+bx+c
&=
(px+q)(rx+s).
\end{aligned}
\]
After expansion, the coefficients must match:
\[
\begin{aligned}
pr&=a,\\
ps+qr&=b,\\
qs&=c.
\end{aligned}
\]
5. Worked example: \(2x^2+8x-24\)
First take out the common factor \(2\):
\[
\begin{aligned}
2x^2+8x-24
&=
2(x^2+4x-12).
\end{aligned}
\]
Now factor the quadratic \(x^2+4x-12\). We need two numbers whose product is \(-12\)
and whose sum is \(4\). Those numbers are \(6\) and \(-2\).
\[
\begin{aligned}
x^2+4x-12
&=
(x+6)(x-2).
\end{aligned}
\]
Therefore:
\[
\begin{aligned}
\boxed{
2x^2+8x-24
=
2(x+6)(x-2)
}.
\end{aligned}
\]
6. Difference of squares
A difference of squares has the form:
\[
\begin{aligned}
a^2-b^2.
\end{aligned}
\]
It factors as:
\[
\begin{aligned}
a^2-b^2
&=
(a-b)(a+b).
\end{aligned}
\]
Example:
\[
\begin{aligned}
x^2-49
&=
x^2-7^2\\
&=
(x-7)(x+7).
\end{aligned}
\]
7. Factoring by grouping
Grouping is useful when a polynomial has four terms and pairs of terms share 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 key is that both groups contain the same factor, \(x+2\).
8. Checking a factorization
To check a factored answer, expand it. If the expansion equals the simplified polynomial,
the factorization is correct.
\[
\begin{aligned}
2(x+6)(x-2)
&=
2(x^2+4x-12)\\
&=
2x^2+8x-24.
\end{aligned}
\]
9. When an expression cannot be factored further
Some polynomials do not factor nicely over the integers.
For example:
\[
\begin{aligned}
x^2+1
\end{aligned}
\]
has no integer binomial factorization. In that case, the expression is usually left as it is.
10. Formula summary
The table below uses plain text formulas in table cells to avoid raw LaTeX rendering problems.
11. Common mistakes
- Forgetting to factor out the GCF before factoring a quadratic.
- Using difference of squares on a sum such as \(x^2+9\).
- Factoring \(x^2+4x-12\) with numbers whose product or sum does not match.
- Dropping a negative sign when expanding or factoring.
- Not checking the answer by expanding the factors.
- Trying to use integer factoring when the polynomial is irreducible over the integers.
Key idea: simplify first, then factor by looking for a common factor, a special pattern, or a quadratic structure.
