How To Factor Polynomials With Coefficients

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.

Step 1

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).

Step 2

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.

Step 3

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.

Step 4

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.