Students typically learn about special products of polynomials during a high school algebra course, as part of a unit on expanding or factoring. There are three primary types of polynomial special products. Unlike most topics in algebra, understanding these three special products of polynomials is optional, because special products can always be multiplied or factored using standard methods.
Advantages of Special Products
Knowing the forms for special products of polynomials saves time when expanding or factoring expressions. Factoring in particular can be a tedious, drawn-out process, sometimes involving a substantial amount of guess-and-check in order to find factors. But by using special products, you can bypass the factoring process entirely, arriving directly at the solution.
Perfect square special products are trinomials, which factor into a squared binomial. They come in two basic forms. Addition perfect square trinomials take the form a^2 + 2ab + b^2, factoring into (a + b)^2, where "a" and "b" are whole numbers. Subtraction perfect square trinomials take the form a^2 -- 2ab + b^2, factoring into (a -- b)^2. Examples of perfect squares can be as basic as x^2 -- 6x + 9, which factors into (x -- 3)^2, or as complex as 4x^2 + 20xy + 25y, which factors into (2x + 5y)^2. You can recognize a perfect square trinomial by the coefficients of its first and last terms, which are both themselves perfect squares.
Difference of Squares
Difference of squares special products consist of a binomial that factors into two binomials. They take the form a^2 -- b^2 and factor into (a -- b)(a + b). For instance, 9x^2 -- 49, which factors to (3x -- 7)(3x + 7), is a difference of squares special product, as is x^2 -- 25xy, which factors to (x -- 5y)(x +5y). The coefficients of both terms in the unfactored binomial are perfect squares.
Sum and Difference of Cubes
The lengthiest of the polynomial special products are the sums and differences of cubes. In their unfactored forms, both are binomials, appearing in a manner similar to the difference of squares special products. A sum of cubes polynomial comes in the form a^3 + b^3, and a difference of cubes polynomial takes the form a^3 -- b^3. The sum of cubes binomial factors to (a + b)(a^2 -- ab + b^2), and the difference of cubes factors into (a -- b)(a^2 + ab + b^2). (See Reference 1) An example of a sum of cubes polynomial is 27x^3 + 8, which factors to (3x + 2)(9x^2 -- 6x + 4). An example of a difference of cubes polynomial is x^3 -- 1, which factors to (x -- 1)(x^2 + x + 1). Both terms in the unfactored sum and difference of cubes binomials are cubes.