A polynomial is a mathematical expression involving one or more terms. Terms are separated by addition or subtraction, and each term may contain coefficients, variables, and exponents that are multiplied together. For example, the following polynomial has three terms:
In the first term, 7 is the coefficient, y is the base, and 4 is the exponent. The coefficients and variables of a term are factors of that term. An expression can also be a factor of a polynomial if it is multiplied by another expression to get the original polynomial as a product.
Distribution involves multiplying a factor outside the parentheses by all the terms inside the parentheses.
Factoring, on the other hand, is the reverse process of distributing, in which a common factor is factored out of two or more terms, and written outside of the parentheses.
Being able to factor a polynomial is useful for:
The process of finding common factors of a polynomial works in the same manner as finding common factors of numbers. To “factor” means to break down a number into smaller numbers that, when multiplied together, give you the original number.
Suppose you want to factor the following polynomial. To do this, you need to factor or break down each individual term into its prime components.
Looking at these three factorizations, you can see that the common factor of each of these terms is 3. If terms have two or more common factors, the factors can be multiplied together to find the greatest common factor.
Here is another example. Again, to factor this polynomial, you need to find the prime factorization of each of the three terms.
You can see that all of your terms have two common factors. They each have a positive 2, and they each have a variable x. To find your greatest common factor, you multiply these two common factors, 2 and x. Therefore, your greatest common factor is 2x.
Once you have factored out the greatest common factor, the expression inside the parentheses will be the remaining factors of each term. From your first term, your remaining factors are 5x and x or 5x^2. In your second term, the remaining factors are a -2 and x, or -2x. Finally, in your last term, the remaining factor is 2.
You can check to see that you have factored the polynomial correctly by distributing the greatest common factor back into each term inside the parentheses:
Because you have arrived back at your original expression, you have factored your polynomial correctly.
Source: This work is adapted from Sophia author Colleen Atakpu.