Begin with a common factor
Look for the greatest common divisor of all integer coefficients. If every term also contains x, factor out the smallest shared power of x. This makes any remaining pattern easier to recognize.
For a monic quadratic x² + bx + c, look for two numbers whose sum is b and whose product is c. Their signs matter: a negative product requires opposite signs.
Be precise about which factors you seek
Some quadratics do not split into rational linear factors even though they have irrational or complex roots. “No rational linear factors” describes a particular kind of factorization, not a failure of algebra.
The linked tool supports integer coefficients through degree two within its bounds. Higher-degree identities can still be studied by hand, but are not covered by that tool’s input contract.
A product becomes an equation only with a condition
Factoring x² − 16 gives (x − 4)(x + 4). That is an identity between expressions, not a declaration that x equals 4 or −4. Those values become solutions only when the original polynomial is set equal to zero. State whether the task asks for factors or roots before presenting the final answer.
LET’S WORK IT THROUGH
Factor 3x² − 12x + 9.
- Extract 3: 3(x² − 4x + 3).
- Find numbers adding to −4 and multiplying to 3: −1 and −3.
- Write the two linear factors.
THE ANSWER3(x − 1)(x − 3).
YOUR TURN
A small check for understanding.
Factor x² − 16.
Reveal the answer
(x − 4)(x + 4), a difference of squares.
Now try your own numbers.
The factor a polynomial includes input guidance and the limits of its supported calculations. Use it alongside the method above.
Open Factor a polynomial ↗A note on precision: preserve exact fractions where possible and round numerical results at the end. Found an unclear step? Prepare a correction report.