Techniques for Factoring Polynomials with Four Terms

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.

Step 1

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.

Step 2

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.

Step 3

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.

Step 4

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).

Step 5

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.

Step 6

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.