How to Factor Polynomials with Coefficients

Factoring polynomials with coefficients is a common problem in precalculus courses.
••• Jupiterimages/Goodshoot/Getty Images

A polynomial is a mathematical expression that consists of variables and coefficients constructed together using basic arithmetic operations, such as multiplication and addition. An example of a polynomial is the expression x^3 - 20x^2 + 100x. The process of factoring a polynomial means simplifying a polynomial into the simplest form that makes the statement true. The problem of factoring polynomials frequently arises in precalculus courses, but performing this operation with coefficients can be completed in a few short steps.

    Remove any common factors from the polynomial, if possible. As an example, the terms in the polynomial x^3 - 20x^2 + 100x have the common factor 'x'. Therefore, the polynomial can be simplified to x(x^2 - 20x + 100).

    Determine the form of the terms that remain to be factored. In the example above, the term x^2 - 20x + 100 is a quadratic with a leading coefficient of 1 (that is, the number in front of the highest power variable, which is x^2, is 1), and therefore can be solved using a specific method to solve problems of this type.

    Factor the remaining terms. The polynomial x^2 - 20x + 100 can be factored into the form x^2 + (a+b)x + ab, which can also be written as (x - a)(x - b), where 'a' and 'b' are numbers that are to be determined. Therefore, the factors are found by determining two numbers 'a' and 'b' that add up to -20 and equal 100 when multiplied together. Two such numbers are -10 and -10. The factored form of this polynomial is then (x - 10)(x - 10), or (x - 10)^2.

    Write the fully factored form of the full polynomial, including all terms that were factored. Concluding the example above, the polynomial x^3 - 20x^2 + 100x was first factored by factoring 'x', giving x(x^2 - 20x +100), and factoring the polynomial within the brackets gives x(x - 10)^2, which is the fully factored form of the polynomial.


About the Author

Thomas Bourdin began writing professionally in 2010. He writes for various websites, where his interests include science, computers and music. He holds a Bachelor of Science degree in physics with a minor in mathematics from the University of Saskatchewan and a Master of Science in physics from Ryerson University.

Photo Credits

  • Jupiterimages/Goodshoot/Getty Images