How To Factor Binomials With Exponents

A binomial is an algebraic expression with two terms. It may contain one or more variables and a constant. When factoring a binomial, you will commonly be able to factor out a single common term, resulting in a monomial times the reduced binomial. If, however, your binomial is a special expression, called a difference of squares, then your factors will be two smaller termed binomials. Factoring simply takes practice. Once you have factored dozens of binomials, you'll more readily see the patterns in them.

Step 1

Make sure you really have a binomial. Look to see if the two terms can be combined into a single term. If each term has the same variable(s) to the same degree, then these can be combined and what you really have is a monomial.

Step 2

Pull out common terms. If both of your terms in the binomial share a common variable(s) then this variable term can be pulled out, or factored out, of each. Pull it out to the degree of the smaller term. For example, if you have 12x^5 + 8x^3 then you can factor out 4x^3. The 4 factors out as the greatest common factor between 12 and 8. The x^3 can factor out because it is the degree of the smaller, common x term. This gives you a factoring of: 4x^3(3x^2 + 2).

Step 3

Check for a difference of squares. If your two terms are each a perfect square and one term is negative while the other is positive, you have a difference of squares. Examples include: 4x^2 – 16, x^2 – y^2, and -9 + x^2. Note in the last, if you switched the order of terms, you would have x^2 – 9. Factor a difference of squares as the square roots of each term added and subtracted. So, x^2 – y^2 factors into (x+y)(x-y). The same holds true with constants: 4x^2 – 16 factors into (2x^2 + 4)(2x^2 – 4).

Step 4

Check if both terms are perfect cubes. If you have a difference of cubes, x^3 – y^3 then the binomial will factor into this pattern: (x-y)(x^2 + xy + y^2). If, however, you have a sum of cubes, x^3 + y^3, then your binomial will factor into (x+y)(x^2 – xy + y^2).