Overview
A factor is an even divisor of another number or expression. For example, 3 is a factor of 12 because 12 can be evenly be divided by 3. The number 4 is a factor of 4x + 8y, because 4 is an even divisor of both 4x and 8y. Polynomials, like other numbers, variables, and expressions, can have factors if they can be evenly divided by another number or expression.
Types of Factors
Some of the types of factors of polynomials include real numbers, monomials with variables, or binomials. Factors are useful to put polynomials in the simplest form, and can also provide shortcuts to solving variables. Suppose the polynomial equation is 3x3 – 6x2 + 9x = 0. It can be factored as 3x(x2 -2x + 3) or 3x(x – 3) (x + 1), so that x can equal 0, 3, or -1.
Figure 1: Factors of polynomials can include real numbers, monomials, and binomials.
Monomials With or Without Variables
Sometimes the easiest factors to find are common factors for each term of the polynomial. Sometimes the variables can be factored out, but sometimes they cannot. Suppose the polynomial expression is 5x2 + 10x -20. It can be factored as 5(x2 + 2x – 4) because 5 times x2 is 5x2, 5 times 2x is 10x and 5 times -4 is -20. Similarly, 64x3 -16x2 + 32x can be factored as 8x(8x2-2x + 4).
Binomials as Factors
Polynomials that can be factored into binomials follow a certain pattern. For example, perfect square trinomials, such as x2 + 18x + 81, can be factored as (x + 9)(x + 9) because the coefficients follow the pattern a2 + 2ab + b2. Similarly, polynomials such as x2 – 25 can be factored as (x + 5) (x – 5). In other polynomials with x2, the squared term has a coefficient of 1 and numbers can be found that their sum can be the middle term and their product can be the last term.
Figure 2: Perfect square trinomials and the difference of squares are easily-factored polynomials.
Try a Special Case
Once a polynomial has been factored, there are two different ways to check to see if the factoring is correct. One way is to multiply the factoring to see if the expressions are equivalent. Another way is to substitute values for the variable to see if the special case works. Suppose that x2 +14x + 49 has been factored as (x +7)(x +7). Substituting 5 for x the first expression can be solved as 25 + 70 + 49 = 144, which equals 122.
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