48 Greatest Common Factor and Factoring

Greatest Common Factor and Factoring is the study of rewriting a polynomial as a product of simpler expressions, beginning with the most fundamental factoring technique — identifying and extracting the largest monomial factor shared by every term — and extending to factoring polynomials with four terms by grouping.

The Scope of Common-Factor Factoring

Factoring reverses the distributive property, rewriting a sum of terms as a product of a common factor and a remaining polynomial. The greatest common factor (GCF) of a polynomial's terms is the largest monomial that divides every term evenly, combining the largest shared numerical factor with the highest power of each variable common to all terms. Extracting the GCF is typically the first factoring technique attempted on any polynomial, since doing so first often simplifies the remaining expression enough to reveal further factoring opportunities.

Finding the GCF of the Coefficients

The numerical portion of a polynomial's GCF is found using the same greatest common factor procedure applied to ordinary integers: identifying the largest number that divides every coefficient in the polynomial without a remainder. For the coefficients 12, 18, and 30, the greatest common factor is 6, since 6 divides each of these numbers evenly and no larger number does.

Finding the GCF of the Variable Factors

The variable portion of a polynomial's GCF is found by identifying every variable common to all terms and taking the lowest exponent that variable carries across those terms. For the terms 8x³y², 12x²y⁴, and 4x⁴y³, the variable x appears in every term with exponents 3, 2, and 4, so its contribution to the GCF is x², using the smallest of those exponents; similarly, y appears with exponents 2, 4, and 3, contributing y² to the GCF.

Constructing the Full Monomial GCF

Combining the numerical GCF with the variable GCF produces the polynomial's complete monomial greatest common factor. For the terms 8x³y² , 12x²y⁴, and 4x⁴y³, the numerical GCF is 4 and the variable GCF is x²y², producing an overall GCF of 4x²y². This constructed GCF is the largest single monomial that divides every term of the polynomial evenly, checked by confirming that each original term divided by the GCF leaves no remainder and no negative exponent.

Factoring Out the GCF

Once the GCF is identified, factoring it out means dividing every term of the polynomial by the GCF and writing the polynomial as the GCF multiplied by the resulting sum of quotients, enclosed in parentheses.

Extracting a Negative Common Factor

When the leading term of a polynomial is negative, or when factoring out a negative version of the GCF produces a more convenient remaining polynomial, a negative common factor may be extracted instead of a positive one, requiring every term inside the resulting parentheses to have its sign reversed relative to the original polynomial.

Factoring by Grouping

When a polynomial has four terms and no single monomial factor is shared by all four, factoring by grouping is often applicable: the four terms are separated into two pairs, a GCF is factored out of each pair independently, and if the two resulting parenthetical expressions match exactly, that shared binomial can itself be factored out as a common factor of the whole expression.

If the two grouped expressions do not match after the initial pairing, rearranging the four terms into a different pairing, or factoring a negative GCF from one of the pairs, is often necessary before the shared binomial factor becomes visible.

Verifying a Factored Polynomial

A factored polynomial is verified by multiplying the extracted factor back across the remaining parenthetical expression, using the distributive property or FOIL as appropriate, and confirming the result matches the original, unfactored polynomial exactly, term for term.

Diagnosing Errors in GCF and Factoring by Grouping

Common errors in this area include identifying only part of the true GCF, such as extracting a coefficient's GCF while overlooking a variable factor common to every term, forgetting to include a factored-out term of 1 when a term of the original polynomial exactly equals the GCF itself, mishandling signs when extracting a negative common factor, and abandoning the grouping method after the first pairing attempt fails to produce matching binomials rather than trying an alternative pairing of the four terms.