Overview
In order to factor polynomials, common factors must be found for every monomial in the expression. It is the reverse process of multiplying polynomials.
Factoring Monomials
When factoring a number, find all factors that have the product of that number. For example, 30 has the factors 1, 30, 3, 10, 2, 15, 5, and 6. The number 36 has the factors 1, 36, 2, 18, 6, and 6. Similarly, when factoring a monomial, find all factors that have the product of that number. For example, 20x2 has the factors 20, x2, 4x, 5x, 20x, x, 2x, 10x and many others.
Negative Factors
Since multiplying a negative number by a negative number results in a positive number, each factor also has corresponding negative numbers as factors, so that 20x2 also has negative factors -20x, -x, -4x, -5x, and so on. It is very important to watch the signs of the expression when factoring, and remember the negative factors.
Polynomial Expressions
Polynomial expressions may be binomials such as 3x2 +12, trinomials such as 4x3 +8z +12q2, or polynomials such as 12m4n4 +3m3n2 +6m2n2. In order for a polynomial expression to be factored, each element of that expression must have the same factor in common. For example, 3x2 and 12 both have a common factor of 3, because 3·x2 is 3x2 and 3·4 is 12. Suppose the expression were 4x3 -3y +5. None of the monomials in that expression have a common factor.
Factoring Polynomials
In order to factor a polynomial, find the common factor of each of the terms in that polynomial. Then factor out that term. The polynomial 4x3 +8z +12q2 has common factors for the coefficients but not for the variables, so that it can be factored out as 2(2x3) + 2(4z) + 2(6q2) or 2(2x3 +4z +6q2). The polynomial 3x4 -8x2y -5x2 has common factors for the variables but not for the coefficients, so that it can be factored as x2(3x2) +x2(-8y) +x2(-5) or x2(3x2-8y-5). The polynomial 12m4n4 +3m3n2 +6m2n2 has common factors in both the variables and the coefficients, so that 3m2n2 (4m2n2) + 3m2n2 (m) + 3m2n2 (2), or 3m2n2 (4m2n2 +m +2). Use the common factor for each term that has the greatest power as well as the largest coefficient.
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