Polynomials are expressions of one or more terms. A term is a combination of a constant and variables. Factoring is the reverse of multiplication because it expresses the polynomial as a product of two or more polynomials. A polynomial of four terms, known as a quadrinomial, can be factored by grouping it into two binomials, which are polynomials of two terms.
Identify and remove the greatest common factor, which is common to each term in the polynomial. For example, the greatest common factor for the polynomial 5x^2 + 10x is 5x. Removing 5x from each term in the polynomial leaves x + 2, and so the original equation factors to 5x(x + 2). Consider the quadrinomial 9x^5 - 9x^4 + 15x^3 - 15x^2. By inspection, one of the common terms is 3 and the other is x^2, which means that the greatest common factor is 3x^2. Removing it from the polynomial leaves the quadrinomial, 3x^3 - 3x^2 + 5x - 5.
Rearrange the polynomial in standard form, meaning in descending powers of the variables. In the example, the polynomial 3x^3 - 3x^2 + 5x - 5 is already in standard form.
Group the quadrinomial into two groups of binomials. In the example, the quadrinomial 3x^3 - 3x^2 + 5x - 5 can be written as the binomials 3x^3 - 3x^2 and 5x - 5.
Find the greatest common factor for each binomial. In the example, the greatest common factor for 3x^3 - 3x is 3x, and for 5x - 5, it is 5. So the quadrinomial 3x^3 - 3x^2 + 5x - 5 can be rewritten as 3x(x - 1) + 5(x - 1).
Factor out the greatest common binomial in the remaining expression. In the example, the binomial x - 1 can be factored out to leave 3x + 5 as the remaining binomial factor. Therefore, 3x^3 - 3x^2 + 5x - 5 factors to (3x + 5)(x - 1). These binomials cannot be factored any further.
Check your answer by multiplying the factors. The result should be the original polynomial. To conclude the example, the product of 3x + 5 and x - 1 is indeed 3x^3 - 3x^2 + 5x - 5.
About the Author
Based in Ottawa, Canada, Chirantan Basu has been writing since 1995. His work has appeared in various publications and he has performed financial editing at a Wall Street firm. Basu holds a Bachelor of Engineering from Memorial University of Newfoundland, a Master of Business Administration from the University of Ottawa and holds the Canadian Investment Manager designation from the Canadian Securities Institute.