Factoring Polynomials
Factoring is the process of rewriting an expression as a product of simpler expressions. It is the reverse of expanding brackets. Mastering factoring is essential for solving quadratic equations, simplifying fractions, and much of advanced algebra.
In this lesson
1 What Factoring Means
Factoring an expression means rewriting it as a multiplication. Just as 12 = 3 × 4, the polynomial x² + 5x + 6 = (x + 2)(x + 3). The factors (x + 2) and (x + 3) multiply together to give the original expression.
Why bother? Because factored form makes it trivial to find where an expression equals zero. If (x+2)(x+3) = 0, you immediately know x = -2 or x = -3. That same conclusion from x² + 5x + 6 = 0 requires more work. (where the expression equals zero), simplify fractions, and solve equations. If (x + 2)(x + 3) = 0, then x = −2 or x = −3 , immediate from factored form, but not obvious from x² + 5x + 6 = 0.
Expanding: (x + 2)(x + 3) → x² + 5x + 6. Factoring: x² + 5x + 6 → (x + 2)(x + 3). They are inverse operations. FOIL (First, Outer, Inner, Last) is used to expand; factoring reverses this process.
2 Step 1: Always Check for GCF First
Before anything else, always check for a common factor. If all terms share one, factor it out first. This simplifies everything that comes after and is the most overlooked step. , this simplifies everything that follows.
3 Difference of Squares
When you see two perfect squares being subtracted, you can factor immediately using a² - b² = (a+b)(a-b). This one is worth recognizing on sight. It applies whenever you have two perfect squares being subtracted.
a² + b² does NOT factor over the real numbers. x² + 25 cannot be factored using real numbers. Only the DIFFERENCE of squares factors this way.
4 Factoring Trinomials
For x² + bx + c (leading coefficient 1): find two numbers that multiply to c and add to b. These numbers become the constants in the factors.
5 Factoring by Grouping
For four-term polynomials, group terms in pairs, factor each pair, then factor out the common binomial.
Try the Quadratic Formula Calculator
Once a polynomial is factored, use the quadratic formula calculator to verify the roots.
Practice Problems
Sources & Further Reading
The explanations on this page draw on the following established sources. We link to primary and secondary sources so you can verify claims and go deeper on any topic.