By Thomas Bourdin, Updated Aug 30, 2022
A polynomial is an algebraic expression made up of variables and coefficients combined with operations like addition and multiplication. For instance, x³ – 20x² + 100x is a typical polynomial. Factoring rewrites the expression into its simplest constituent factors while preserving equality. Although factoring is a common topic in precalculus, it can be tackled efficiently—even when coefficients are involved—by following a few systematic steps.
In the example x³ – 20x² + 100x, every term is divisible by x, so we factor out x to obtain x(x² – 20x + 100).
The bracketed part x² – 20x + 100 is a monic quadratic (leading coefficient 1), which allows us to use the standard factoring technique for quadratics.
We seek two numbers that add to –20 and multiply to 100. The pair –10 and –10 satisfies both conditions, giving (x – 10)(x – 10) or (x – 10)².
Incorporating the extracted common factor, the complete factorization is x(x – 10)².
By following these four steps, any polynomial with integer coefficients can be systematically factored.
