Why Factor Out a Common Factor?
A polynomial may be written as a sum of terms, but sometimes each term contains the same numerical or variable factor. Rewriting the sum as a product by taking out that shared factor is called factoring out the greatest common factor, or GCF. It is a basic way to reveal the structure of a polynomial and to express an equivalent formula in a more useful form.
The operation reverses the distributive property. For example, the distributive property gives \(6x+12=6(x+2)\). Factoring recognizes that both terms on the left contain 6. The same idea applies when terms contain variables, several variables, or negative coefficients.
To factor correctly, identify what is shared by every term. A factor shared by only some terms cannot be taken outside the entire expression. After factoring, distribute the proposed factor back through the parentheses to check that every original term is recovered.
Finding the Greatest Common Factor
A polynomial is a sum of terms. In an expression such as \(12x^3y^2-18x^2y^3+6xy\), the terms are \(12x^3y^2\), \(-18x^2y^3\), and \(6xy\). The coefficient of a term is its numerical multiplier; the variable part records the variables and their exponents.
For example, the coefficients \(12\), \(-18\), and \(6\) have greatest positive common divisor 6. The variable \(x\) appears in all three terms with exponents 3, 2, and 1, so the smallest exponent is 1. The variable \(y\) appears in all three terms with exponents 2, 3, and 1, so its smallest exponent is also 1. Thus their GCF is \(6xy\).
The sign of a coefficient does not change which positive integer divides it. In particular, the GCF is conventionally taken to be positive. A negative common factor can also be factored out, but it is not the positive GCF; sometimes it is chosen to make the first term inside the parentheses positive.
A Reliable Procedure
Work term by term rather than relying on a visual guess. First find the greatest positive common divisor of all the numerical coefficients. Then inspect each variable: include it only if it occurs in every term, and use the smallest exponent with which it occurs. Finally, divide each original term by the resulting GCF, keeping all signs. The quotients become the terms inside the parentheses.
Keep the sign of each coefficient when forming the quotients.
Choose the largest positive integer that divides every coefficient.
Include only variables present in every term, each to its smallest exponent.
Write the quotients as a sum or difference inside parentheses.
Multiplying the factor through the parentheses must reproduce the original expression exactly.
Why the Procedure Works
The justification is the distributive property. Suppose each term of a polynomial can be written as \(gq_i\), where \(g\) is the chosen common factor and \(q_i\) is the quotient for the \(i\)-th term. The sum of the terms can then be rewritten by factoring out \(g\).
Distributing the right-hand side returns the expression on the left. This proves that factoring out a common factor preserves the value of the expression. The “greatest” part of the GCF tells us that we have removed as much common numerical and variable content as possible under the definition; it is not required just to make an equivalent factorization, but it gives a standard, maximally shared factor.
For the variable part, the smallest exponent is exactly what can be removed from every term. If \(x\) occurs as \(x^4\), \(x^2\), and \(x^3\), then \(x^2\) divides each power, but \(x^3\) does not divide \(x^2\). Thus the common power is \(x^2\). The same reasoning applies independently to each variable.
Worked Examples
Worked Example: Factoring a Positive Leading Term with Two Variables
Factor the GCF from \(12x^3y^2-18x^2y^3+6xy\).
The numerical coefficients are \(12\), \(-18\), and \(6\). Their greatest positive common divisor is 6. Each term contains \(x\), and the smallest exponent of \(x\) is 1. Each term also contains \(y\), and the smallest exponent of \(y\) is 1. The GCF is therefore \(6xy\).
Divide each term by \(6xy\), retaining its sign:
Write those quotients inside parentheses:
Check by distributing: \(6xy(2x^2y)=12x^3y^2\), \(6xy(-3xy^2)=-18x^2y^3\), and \(6xy(1)=6xy\). Adding these products gives the original expression, so the factorization is correct.
Worked Example: Factoring When the Leading Term Is Negative
Factor the GCF from \(-15x^4+25x^3-10x^2\), and choose a negative factor so the first term in parentheses is positive.
The positive numerical GCF of \(15\), \(25\), and \(10\) is 5. Every term contains \(x\), and the smallest exponent is 2. Thus the positive GCF is \(5x^2\). Because the leading coefficient is negative, factor out \(-5x^2\) instead. Divide each term by this negative factor:
Therefore,
Check by distributing: \((-5x^2)(3x^2)=-15x^4\), \((-5x^2)(-5x)=25x^3\), and \((-5x^2)(2)=-10x^2\). The original expression is recovered. Factoring out \(5x^2\) instead would also be valid, giving \(5x^2(-3x^2+5x-2)\); the negative choice simply makes the first term inside the parentheses positive.
Worked Example: Do Not Include a Variable That Is Not in Every Term
Factor the GCF from \(14a^3b^2-21a^2b+7a^2\).
The greatest positive common divisor of \(14\), \(-21\), and \(7\) is 7. The variable \(a\) occurs in all three terms; its exponents are 3, 2, and 2, so the smallest is 2. The variable \(b\) does not occur in the last term, so it is not common to all terms. The GCF is \(7a^2\).
Divide each term by \(7a^2\):
Hence,
Check: \(7a^2(2ab^2)=14a^3b^2\), \(7a^2(-3b)=-21a^2b\), and \(7a^2(1)=7a^2\). These are exactly the three original terms.
Common Mistakes and How to Correct Them
A frequent error is to take out a variable that appears in most terms but not all. For instance, in \(8x^2+12x\), both terms contain \(x\), so it can be factored out. But in \(8x^2+12\), the constant term has no \(x\); \(x\) is not a common factor of the entire expression.
Another error is to divide coefficients but forget to divide each term by the complete GCF, or to lose a negative sign when forming a quotient. Consider the following incorrect work:
Distributing the proposed factor gives \(6x(2x-3)=12x^2-18x\), which has the opposite sign from the original polynomial. The positive GCF is \(6x\), but dividing the negative first term by \(6x\) gives \(-2x\), not \(2x\). Correctly using the positive GCF gives
Alternatively, take out \(-6x\) to make the first term inside positive:
Distributing the latter gives \(-12x^2+18x\), confirming the signs. A quick check by multiplication catches this kind of error before it carries into later algebra.
How to Recognize a Complete GCF Factorization
A factorization by GCF is complete when the factor outside the parentheses is the greatest common factor of the original terms. Check both components: the positive numerical factor should be as large as possible, and each variable outside should occur to the smallest exponent found across all terms. The expression inside may still have other algebraic structure, but identifying or factoring that structure is a separate task.
For example, if a factorization leaves a numerical factor that divides every term inside the parentheses, then the original numerical factor was not greatest. Similarly, if every term inside still contains \(x\), an additional \(x\) could have been included in the factor outside. These observations provide a useful completeness check without changing the essential multiplication check.
Check Your Understanding
For each question, identify the common factor carefully and retain the signs of the original terms.
- What is the greatest common factor of \(20x^4y^2\) and \(30x^2y^3\)?
- Factor the GCF from \(18m^3-24m^2\).
- Factor the GCF from \(-14p^3+21p^2\), choosing a negative outside factor.
- Why is \(a\) not part of the GCF of \(6a^2b+9b\)?
- How can distributing the proposed factor check whether a factorization is correct?
