If an expression cannot be factored it is said to be prime. Remember, this is a check to make sure we have factored correctly.Īgain, find the greatest common factor of the numbers and each letter separately. Then, "What is the largest common factor of x 3, x 2, and x?" Say to yourself, "What is the largest common factor of 12, 6, and 18?" Multiplying, we get the original and can see that the terms within the parentheses have no other common factor, so we know the solution is correct. For instance, 6 is a factor of 12, 6, and 18, and x is a factor of each term. In other words, don’t attempt to obtain all common factors at once but get first the number, then each letter involved. A good procedure to follow is to think of the elements individually. You should be able to mentally determine the greatest common factor.
FThe expression must be completely factored.Īt this point it should not be necessary to list the factors.It must be possible to multiply the factored expression and get the original expression.This expression is factored but not completely factored.įor factoring to be correct the solution must meet two criteria: Hence, the expression is not completely factored. However, the factor x is still present in all terms. Multiplying to check, we find the answer is actually equal to the original expression. If we had only removed the factor "3" from 3x 2 + 6xy + 9xy 2, the answer would be In other words, "Did we remove all common factors? Can we factor further?" A second check is also necessary for factoring - we must be sure that the expression has been completely factored. If the answer is correct, it must be true that. To check the factoring keep in mind that factoring changes the form but not the value of an expression. The original expression is now changed to factored form. It is the reverse of the process that we have been using until now. Note that this is the distributive property. The terms within the parentheses are found by dividing each term of the original expression by 3x. Proceed by placing 3x before a set of parentheses. In this case, the greatest common factor is 3x. Next look for factors that are common to all terms, and search out the greatest of these. To factor an expression by removing common factors proceed as in example 1.ģx is the greatest common factor of all three terms. Each term of 10x + 5 has 5 as a factor, and 10x + 5 = 5(2x + 1). In general, factoring will "undo" multiplication. In the previous chapter we multiplied an expression such as 5(2x + 1) to obtain 10x + 5. Determine which factors are common to all terms in an expression.Upon completing this section you should be able to: Note that in this definition it is implied that the value of the expression is not changed - only its form. Factors can be made up of terms and terms can contain factors, but factored form must conform to the definition above.įactoring is a process of changing an expression from a sum or difference of terms to a product of factors. Note in these examples that we must always regard the entire expression. Factors occur in an indicated product.Īn expression is in factored form only if the entire expression is an indicated product. Terms occur in an indicated sum or difference. You should remember that terms are added or subtracted and factors are multiplied.
In earlier chapters the distinction between terms and factors has been stressed. In fact, the process of factoring is so important that very little of algebra beyond this point can be accomplished without understanding it. Consider \(x=20\) miles per hour to be the only solution.The process of factoring is essential to the simplification of many algebraic expressions and is a useful tool in solving higher degree equations. The negative answer does not make sense in the context of this problem.
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