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