Algebra: Factoring

Throughout math, you will use a process known as factoring in many different problems. It is used when solving polynomial equations, to simplify things, and for many other purposes.

This page will help you better understand how to factor and how to find a GCF, or Greatest Common Factor. Follow any of the links below to be on your way to a better understanding of those concepts, and good luck!

GCF, or Greatest Common Factor
The process known as factoring
Factoring trinomials
Quiz on factoring

Greatest Common Factor

If you know nothing about finding a greatest common factor, a good place to start would be at the Pre-Algebra page on fractions and finding a GCF.

In Algebra, the greatest common factor is found a little bit differently than it is in Pre-Algebra. In Algebra, only prime factors of numbers are used, and in many cases, you will be asked to find the GCF of algebraic terms. Following are two examples:

     2 * 3 * 5 * 7 = 210
     210xy^2z^3 = 2 * 3 * 5 * 7 * x * y * y * z * z * z

As mentioned above, only prime numbers and literal factors, the letters, are used in this factoring process. Because only the prime and literal factors are used, the GCF is defined as follows: The GCF of two or more terms is the product of all prime algebraic factors common to every term, each to the highest power that it occurs in all the terms. Putting that in more reasonable terms tells us that the GCF has to be made of factors that are present in all the terms for which you are finding the GCF. Example:

     The expression 6x^2y^2m^2 + 3xy^3m^2 + 3x^3y^2 can be rewritten
     as a product of prime and literal factors --

     2 * 3 * x * x * y * y * m * m + 3 * x * y * y * y * m * m + 3 * x * x * x * y * y.

     Since the first term is the only term with 2 as a factor, 2 is not a
     factor of the GCF.  Each term has 3 as a factor at least once, so 3 is a
     factor of the GCF.

     3

     Each term also has x as a factor at least once, so x is a factor of the
     GCF.

     3x

     y is a factor of each term twice, and m is not a factor of all the
     terms, so it is not a part of the GCF.

     3xy^2 is the GCF.

The Process of Factoring

Using the distributive property lets us change an expression from a product to a sum. For example, an expression such as 3a(x - c) tells you to multiply 3a by x - c. When you do that, you get the sum 3ax - 3ac. When you do that in reverse, by writing 3ax - 3ac as the product of the two factors 3a and x - c, you are factoring.

Example:

1. Factor:   4(a^3)(b^4)(z^3) + 2(a^2)b(z^4)
   Solution: Write out the terms as products of their prime and literal
             factors.

             2*2*a*a*a*b*b*b*b*z*z*z + 2*a*a*b*z*z*z*z

             Each term has at least one 2, two a's, one b,
             and three z's as factors.  Therefore, the GCF is
             2a^2bz^3.

             (2a^2bz^3)(     )

             Now that you've got the GCF factored out, you can rewrite the two
             terms without the factors in the GCF.

             2 * a * b * b * b + z

             The second pair of parentheses can now be filled in with the 
             rewritten terms.

             (2a^2bz^3)(2ab^3 + z) is the answer.

Factoring Trinomials

In this section, we will only help you better understand how to factor quadratic trinomials, or trinomials whose highest power is two. Also, we assume you know how to multiply binomials (we use the "FOIL" method).

Using a multiplication problem consisting of two binomials, we will show some important things to remember when factoring trinomials, which is the reverse of multiplying two binomials. Example:

     (x - 6)(x + 3) = x^2 - 6x + 3x - 18 = x^2 - 3x - 18

     1. The first term of the trinomial is the product of the first two terms
        of the binomials.
     2. The last term of the trinomial is the product of the last terms of the
        binomials.
     3. The coefficient of the middle term of the trinomial is the sum
        of the last terms of the binomials.
     4. If all the signs in the trinomial are positive, all signs in both
        binomials are positive.

Keeping these important things in mind, you can factor trinomials.

Example:

1. Factor:   x^2 - 14x - 15
   Solution: First, write down two sets of parentheses to indicate the
             product.

             (     )(     )

             Since the first term in the trinomial is the product of the first
             terms of the binomials, you enter x as the first term of each
             binomial.

             (x    )(x    )

             The product of the last terms of the binomials must equal -15,
             and their sum must equal -14, and one the binomials' terms has
             to be negative.  Four different pairs of factors have a product
             that equals -15.

             (3)(-5) = -15     (-15)(1) = -15
             (-3)(5) = -15     (15)(-1) = -15

             However, only one of those pairs has a sum of -14.

             (-15) + (1) = -14

             Therefore, the second terms in the binomial are -15 and 1
             because these are the only two factors whose product is -15 (the
             last term of the trinomial) and whose sum is -14 (The
             coefficient of the middle term in the trinomial).

             (x - 15)(x + 1) is the answer.

Take the quiz on factoring. The quiz is very useful for either review or to see if you've really got the topic down.

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