How to Factor Polynomials for Beginners

Factoring higher-order polynomials takes patience and practice.
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Polynomials are groups of mathematical terms. Factoring polynomials allows them to be solved easier. A polynomial is considered factored completely when it is written as a product of the terms. This means no addition, subtraction, or division left behind. By using methods you have learned early on in school, you will be able to factor polynomials. After a little practice, factoring becomes easier and more fun.

Greatest Common Factor Method

    Determine the greatest common factor of the polynomial. This can be absolutely anything every term has in common. For instance, the polynomial 5xy+35y+10y2 has the factor 5y in common. Another example is 5(x+y) – 2x(x+y). This polynomial has (x+y) in common.

    Divide out the greatest common factor. In the above examples, you would have 5y(x+7+2y) and (x+y)(5-2x).

    Check the factors by multiplying them back out. If you reach the original polynomial, then your factors are correct.

Grouping Method

    Group terms together if you have four terms without a greatest common factor.

    Group the first two terms together and the last two terms together. For instance, x3+5x2+2x+10 would be grouped as (x3+5x2)+(2x+10).

    Find the greatest common factor for each group. (x3+5x2)+(2x+4) would become x2(x+5)+2(x+5).

    Factor out the common binomial. In this case that would be (x+5).

    Combine the outer terms into their own factor: (x2+2)(x+5).

    Check the factors by multiplying them back out. If you reach the original polynomial, then your factors are correct.

    Tips

    • Some polynomials cannot be factored using the greatest common factor. These will require synthetic division and sometimes will still not be able to be factored.

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