Factoring Polynomials

Factoring polynomials means rewriting a polynomial as a product. For example, $x^2 - 5x + 6$ factors to $(x - 2)(x - 3)$. The expression is equivalent, but the factored form is often easier to solve, simplify, and interpret.

If you are searching for how to factor polynomials, the core idea is simple: pull out any common factor first, then check whether the remaining expression matches a known pattern.

What Factoring Tells You

Factored form exposes structure that is hidden in expanded form. If

$$x^2 - 5x + 6 = (x - 2)(x - 3),$$

then the zeros are easy to read: $x = 2$ or $x = 3$. That matters when you are solving equations, finding $x$-intercepts, or simplifying rational expressions.

This shortcut depends on the expression actually being written as a product. You cannot read zeros directly from the expanded form alone.

Start With The Greatest Common Factor

Before trying a pattern, check whether every term shares a number, a variable, or both. This is the fastest factoring step, and missing it often makes the rest of the problem harder.

For

$$6x^2 + 9x$$

both terms share $3x$, so factor that out first:

$$6x^2 + 9x = 3x(2x + 3)$$

That is already fully factored over the integers.

Common Patterns To Recognize

Many polynomial factoring problems become manageable once you identify the shape.

Trinomials

For a trinomial such as

$$x^2 + bx + c,$$

look for two numbers that multiply to $c$ and add to $b$. This direct method works when the leading coefficient is $1$.

Difference Of Squares

If you see

$$a^2 - b^2,$$

then

$$(a - b)(a + b).$$

This works because the middle terms cancel when you multiply back out.

Grouping

For a four-term polynomial, grouping can help. It works only if the same binomial factor appears after you factor each pair.

Worked Example: Factor $2x^2 + 7x + 3$

This example shows a trinomial whose leading coefficient is not $1$:

$$2x^2 + 7x + 3.$$

Multiply the leading coefficient and the constant term:

$$2 \cdot 3 = 6.$$

Now find two numbers that multiply to $6$ and add to $7$. Those numbers are $6$ and $1$.

Split the middle term with those two numbers:

$$2x^2 + 7x + 3 = 2x^2 + 6x + x + 3.$$

Group the terms:

$$(2x^2 + 6x) + (x + 3).$$

Factor each group:

$$2x(x + 3) + 1(x + 3).$$

Now the common binomial factor appears:

$$2x^2 + 7x + 3 = (2x + 1)(x + 3).$$

Check by expanding:

$$(2x + 1)(x + 3) = 2x^2 + 6x + x + 3 = 2x^2 + 7x + 3.$$

If you cannot find integer pairs that work in this step, the polynomial may factor differently or may not factor nicely over the integers.

Common Mistakes When Factoring Polynomials

1. Skipping the greatest common factor. For $4x^2 - 8x$, the fully factored form is $4x(x - 2)$, not just $2x(2x - 4)$.
2. Forcing the wrong pattern. For example, $a^2 + b^2$ is not a difference of squares over the real numbers.
3. Losing a sign. One sign error changes the middle term immediately.
4. Forgetting to check. A factorization is only confirmed after expansion gives the original polynomial back exactly.

When You Use Factoring

Factoring is most useful when you need to:

1. Solve polynomial equations
2. Simplify rational expressions
3. Find x-intercepts of polynomial graphs
4. Rewrite expressions before later algebra or calculus steps

The method depends on the polynomial. Some expressions factor cleanly over the integers, some only over larger number systems, and some do not factor into simpler pieces at all.

Try A Similar Problem

Try factoring $x^2 - 9x + 20$. Start by asking which two numbers multiply to $20$ and add to $-9$, then expand your answer to check it.

If you want to compare your steps with another worked solution, try your own version in a solver after you finish the expansion check by hand.