What does factoring an expression mean?
Factoring reverses multiplication or distribution. For example, expanding 3x(x + 2) gives 3x² + 6x, so factoring 3x² + 6x returns 3x(x + 2). The expressions are equivalent for every x, but the product form exposes zeros, common structure, and possible cancellation in later work.
A polynomial is factored completely over a chosen number system when none of its nonconstant factors can be factored further there. This calculator focuses on supported rational polynomial factors; an irreducible result over the rationals may still factor using irrational or complex numbers.
How to factor a polynomial step by step
Always check for a greatest common factor first. Removing it reduces the coefficients and degree pattern, which makes any remaining trinomial or special product easier to recognize.
- Write the polynomial in descending-power standard form and combine like terms.
- Factor out the greatest common numerical and variable factor, including −1 when useful.
- Count the remaining terms and inspect the degree pattern.
- Apply a valid pattern such as a quadratic trinomial or difference of squares.
- Repeat until each factor is irreducible over the selected rational scope.
- Expand the complete product and compare it term by term with the original polynomial.
Essential factoring formulas and patterns
A difference of squares has two squared terms separated by subtraction and factors into conjugates. A perfect-square trinomial has first and last terms that are squares and a middle term equal to twice their product. These patterns should match exactly before they are used.
A sum of squares a² + b² does not use the real-number difference-of-squares pattern. Pattern recognition must include both the term shapes and the operation between them.
Worked example: x² − 5x + 6
For a monic trinomial x² + bx + c, look for two numbers whose product is c and whose sum is b. Here the numbers multiply to 6 and add to −5, so they are −2 and −3.
Therefore x² − 5x + 6 = (x − 2)(x − 3). Verify by expansion: x² − 3x − 2x + 6 = x² − 5x + 6. If the expression were part of the equation x² − 5x + 6 = 0, the zero-product property would then give x = 2 or x = 3.
GCF and non-monic quadratic trinomials
For 6x² + 9x, the GCF is 3x, giving 3x(2x + 3). Leaving out the GCF means the expression is not fully factored. For a non-monic trinomial such as 6x² + 7x + 2, the leading coefficient must be distributed across the factor pair: (3x + 2)(2x + 1).
The product of the first terms must equal the leading term, the product of the last terms must equal the constant, and the outer-plus-inner terms must reproduce the middle coefficient. Expansion is the fastest complete proof.
Common factoring mistakes
Common errors include skipping the GCF, using the difference-of-squares formula on a sum, choosing factor constants with the right product but the wrong sum, and stopping before all factors are complete. Sign pairs deserve special attention when c is negative.
Do not judge a factorization by appearance. Multiply every factor back together and combine like terms. If any coefficient differs from the original, the proposed factorization is not equivalent.