← All math guides · Open the calculator
How to Factor Polynomials
A checklist that works on almost every problem, with an example for each step.
Factoring means rewriting a polynomial as a product of simpler pieces, the way 12 can be written as 2 × 2 × 3. It is the key step in solving equations, simplifying fractions and finding where a graph crosses the x-axis. Work through these methods in order and you will rarely get stuck.
Step 1: Take out the greatest common factor
Always start here. Find the largest number and highest power of x that divides every term, and pull it out front.
Example: Factor 6x³ + 9x².
Both terms are divisible by 3 and by x², so the answer is 3x²(2x + 3).
Try it in CasCalc →Step 2: Look for a difference of squares
Two perfect squares with a minus sign between them always factor the same way:
Example: Factor x² − 49. Since 49 = 7², the answer is (x − 7)(x + 7).
Note that a sum of squares such as x² + 49 does not factor using real numbers.
Step 3: Factor trinomials of the form x² + bx + c
Find two numbers that multiply to c and add to b.
Example: Factor x² + 7x + 12.
3 and 4 multiply to 12 and add to 7, so the answer is (x + 3)(x + 4).
Try it in CasCalc →Signs matter. If c is positive, both numbers have the same sign as b. If c is negative, the numbers have opposite signs.
Step 4: The ac method for ax² + bx + c
When the leading coefficient is not 1, multiply a and c, find two numbers that multiply to ac and add to b, split the middle term, and group.
Example: Factor 6x² + x − 2.
ac = 6 × (−2) = −12. The numbers 4 and −3 multiply to −12 and add to 1.
Split the middle: 6x² + 4x − 3x − 2 = 2x(3x + 2) − 1(3x + 2) = (2x − 1)(3x + 2).
Try it in CasCalc →Step 5: Factor by grouping (four terms)
With four terms, pair them up, factor each pair, and look for a shared bracket.
Example: Factor x³ + 2x² − 4x − 8.
Group: x²(x + 2) − 4(x + 2) = (x + 2)(x² − 4).
The second bracket is a difference of squares, so the full answer is (x + 2)²(x − 2).
Try it in CasCalc →Step 6: Sums and differences of cubes
Examples: x³ − 8 = (x − 2)(x² + 2x + 4) and x³ + 27 = (x + 3)(x² − 3x + 9).
A memory trick for the signs is SOAP: Same sign, Opposite sign, Always Positive.
Check by expanding
Multiply your factors back out. If you get the original polynomial, you are done. In CasCalc, choose Expand on your factored answer to check it in one click.
Why factoring matters
Once a polynomial is factored, solving it is easy: set each factor equal to zero. From (x + 2)²(x − 2) = 0 you can read off the solutions x = −2 and x = 2 immediately. Factoring is also how you simplify algebraic fractions, by cancelling a bracket that appears on the top and the bottom.
CasCalc · Math guides · About · Privacy