Math Review of Finding Common Factors of Polynomials

Overview:

The process of finding common factors for polynomials is similar to the process of finding common factors for real numbers. Both can be written as prime factors.  The greatest common factor may be numerical, the product of a real number and a variable, or a binomial expression.

Finding Prime Factors of Real Numbers

There are two types of real numbers: prime numbers and composite numbers.  Prime numbers such as 2, 3, 5, 7, 11, 13 and so on, are only divisible by themselves and 1.  All other numbers are composite numbers, as the product of prime numbers.  Factoring a real number is a process of determining its prime factors.  Suppose the number is 756.  Its factors can be approximated by dividing by 9 to equal 9 ∙ 84.  Both 9 and 84 are composite numbers, so 9 can be factored as 3²and 84 can be factored as 3∙7∙2².  The prime factorization of 756 is then 2^{2 ∙} 3³ ∙ 7.

Finding Numerical Factors of Polynomials

Finding numerical factors of polynomials is a very similar process.  Suppose the expression is 12x² + 4x + 8.  It can be factored by finding the numerical factor that is common to all monomials in the expression.  In this example, every term can be divided evenly by 4, or 4(3x² + x + 2).  The real number 12 is factored as 2²∙ 3, the real number 4 is factored as 2², and the real number 8 is factored as 2³, or 2² ∙ 2.

Finding Greatest Common Factors that Include Variables

Sometimes the greatest common factor includes both a numerical coefficient and a variable.  Suppose the expression is 15y³+25y² + 10y.  In that case, the polynomial can be factored as the product of a monomial and a polynomial, as 5y(3y² + 5y + 2).  If the polynomial were 18x³y + 21x²y² + 33xy³, it could be factored as 3xy(9x² + 7xy + 11y²).

Finding Greatest Common Factors that Include Binomials

Sometimes the greatest common factors have a binomial factor.  Suppose the problem is by + bd + xy + dx.  Each group can be factored further, as b(y + d) + x(y + d).  Using the Distributive Property results in (b + x)(y + d).

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