A polynomial is an algebraic expression with more than one term. Binomials have two terms, trinomials have three terms and a polynomial is any expression with more than three terms. Factoring is the division of the polynomial terms to their simplest forms. A polynomial is broken down to its prime factors and those factors are written as a product of two binomials, e.g., (x + 1)(x – 1). A greatest common factor (GCF) identifies a factor that all terms within the polynomial have in common. It can be removed from the polynomial to simplify the factoring process.
How to Factor Binomials
Examine the binomial x^2 – 49. Both terms are squared and because this binomial uses the subtraction property, it is called a difference of squares. Note there is no solution for positive binomials, e.g., x^2 + 49.
Find the square roots of x^2 and 49. √X^2 = x and √49 = 7.
Write the factors in parentheses as the product of two binomials, (x + 7)(x – 7). Because the last term, -49, is negative, you will have one of each sign -- because a positive multiplied by a negative equals a negative.
Check your work by distributing the binomials, (x)(x) = x^2 + (x)(-7) = -7x + (7)(x) = 7x + (7)(-7) = -49. Combine like terms and simplify, x^2 + 7x – 7x – 49 = x^2 – 49.
How to Factor Trinomials
Examine the trinomial x^2 – 6xy + 9y^2. Both first and last terms are squares. Because the last term is positive and the middle term is negative, there will be two negative signs within the parenthetical binomials. This is called a perfect square. This term applies to trinomials that have two positive terms as well, x^2 + 6xy + 9y^2.
Find the square roots of x^2 and 9y^2. √x^2 = x and √9y^2 = 3y.
Write the factors as the product of two binomials, (x – 3y)(x – 3y) or (x – 3)^2.
Examine the trinomial x^3 + 2x^2 – 15x. In this trinomial, there is a greatest common factor, x. Pull x from the trinomial, divide the terms by the GCF and write the remainders in parentheses, x(x^2 + 2x – 15).
Write the GCF in front and the square root of x^2 in parentheses, setting up the formula for the product of two binomials, x(x + )(x - ). There will be one of each sign in this formula because the middle term is positive and the last term is negative.
Write down the factors of 15. Because 15 has several factors, this method is called trial-and-error. When looking through the factors of 15, look for two that combine to equal the middle term. Three and five will equal two when subtracted. Because the middle term, 2x is positive, the larger factor will follow the positive sign in the formula.
Write the factors 5 and 3 into the binomial product formula, x(x + 5)(x – 3).
How to Factor Polynomials
Examine the polynomial 25x^3 – 25x^2 – 4xy + 4y.To factor a polynomial with four terms, use a method called grouping.
Separate the polynomial down the center, (25x^3 – 25x^2) – (4xy + 4y). With some polynomials, you may have to rearrange the terms before grouping so that you can pull a GCF out of the group.
Pull the GCF from the first group, divide the terms by the GCF and write the remainders in parentheses, 25x^2(x – 1).
Pull the GCF from the second group, divide the terms, and write the remainders in parentheses, 4y(x – 1). Notice the parenthetical remainders match; this is the key to the grouping method.
Rewrite the polynomial with the new parenthetic groups, 25x^2(x – 1) – 4y(x – 1). The parentheses are now common binomials and can be pulled from the polynomial.
Write the remainder in parentheses, (x – 1)(25x^2 – 4).
TL;DR (Too Long; Didn't Read)
Always redistribute the product of binomials to check your work. Math errors made through factoring are simple, usually incorrect sign arrangements or wrong calculations.