Given the product of a multiplicative expression, factoring is the process of breaking a number down into its smaller factor components. For example, the integer can be factored in several different ways: Similarly, multiplicative algebraic expressions, such as can be rewritten by rewriting their coefficients and variables as a product of their factors.
When all terms in an expression contain a common factor, the expression can be rewritten as a product of its factors. Each term in the expression is divided by the common factor. It is written in front of a parentheses which contains the quotient of the terms. Consider the expression Notice that each term contains Factoring out results in the product Notice that if we use the Distributive Property on the factors, what results is the original expression.
Find and factor the GCF in the following expressions.
To find the GCF between terms, it's necessary to analyze the number and variable part of each term. Notice that means Thus, and are both factors of this term. Additionally, can be written in terms of its factors. This gives The second term of the original expression, does not have any variable part. Thus, it can be written in terms of its factors as Notice that and both share a factor of Thus, is the GCF. Factoring out of the expression gives
Following the same process as above, the terms in this expression can be written in terms of their factors. Notice that each term contains a and an in its factors. Thus, is the expression's GCF. Factoring gives
An expression written in factored form and set equal to can be solved using the Zero Product Property. When the product of two or more factors is at least one of the factors must equal Consider the following equation.
Since one of the factors must equal set them each equal to and solve for the variable. Notice that new equations are created.
seeits factors. This expression can be factored by finding a pair of integers whose product is which here is and whose sum is which in the example is
The first step is to find all possible pairs of integers that multiply to In this case, and Thus, their product is To find all factor pairs, start with the pair where one factor is The other factor must then be Then continue with the pair where one of the factors is and so forth. In this case, there are three pairs.
If the given expression is factorable, one of the factor pairs will add to equal In this case, Here, and is the only factor pair that adds to
Now, use the factor pair to rewrite the -term of the original expression as a sum. Since the factor pair is and the middle term can be written as This gives the following equivalent expression. Notice that the expression hasn't been changed. Rather, it has been rewritten.
Now, the expression has four terms, which can be grouped into the first two terms and the last two terms. Then, the GCF of each group can be factored out. Here, begin with the first group of terms, and Notice how each can be written as a product of its factors. The GCF is Therefore, it's possible to factor out
Next, repeat the same process with the last two terms. In this case, and do not have any common factors, but it's always possible to write expressions as a product of and itself.
Solve the following equation by factoring.
The area of a rectangle is square meters and can described by Write an equation that represents this area. Solve the equation to find