Polynomials are mathematical equations that contain variables and constants. They may also have exponents. The constants and the variables are combined by addition, while each term with the constant and the variable is connected to the other terms by either addition or subtraction. Factoring polynomials is the process of simplifying the expression by division. In order to factor polynomials, you must determine whether it is a binomial or a trinomial, understand the standard factoring formats, find the greatest common factor, find which numbers corresponds to the product and sum of the various parts of the polynomial and then check your answer.
Determine whether the polynomial is a binomial or a trinomial. A binomial has two terms, and a trinomial has three terms. An example of a binomial is 4x-12, and an example of a trinomial is x^2 + 6x + 9.
Understand the difference between the difference of two perfect squares, the sum of two perfect cubes and the difference of two perfect cubes. These types of polynomials are binomials and have a special format for factoring. For example, x^2-y^2 is the difference of two perfect squares. You factor it by finding the square root of each term, subtracting them in one set of parenthesis and adding them in the other, such as (x+y)(x-y). The polynomial x^3-y^3 is the difference of two perfect cubes. After you find the cube root of each term, you put it in the format (x-y)(x^2+xy+y^2). The sum of two perfect cubes is x^3+y^3. The format for factoring that is (x+y)(x^2-xy+y^2).
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Find the greatest common factor. The greatest common factor is the highest number that is divisible by all of the constants in the polynomial. For example, in 4x-12, the greatest common factor is 4. Four divided by four is one, and 12 divided by four is three. By factoring out the four, the expression simplifies to 4(x-3).
Find the numbers which correspond to the product and the sum of the second and third terms of the polynomial. This is how you factor trinomials. For example, in the problem x^2+6x+9, you need to find two numbers that add up to the third term, nine, and two numbers that multiply to the second term, six. The numbers are three and three, as 3 * 3=9 and 3+3=6. The polynomial factors to (x+3)(x+3).
Check your answer. In order to make sure you factored the polynomial correctly, multiply the contents of the answer. For example, for the answer 4(x-3), you would multiply four by x, and then subtract four times three, such as 4x-12. Since 4x-12 is the original polynomial, your answer is correct. For the answer (x+3)(x+3), multiply the x by the x, then add the x times three, then add x times three, and then add three times three, or x^2+3x+3x+9, which simplifies to x^2+6x+9.