Math101 · Independent Ontario learning library
Math101Greatest Common Factor Factoring
Greatest common factor factoring extracts the largest monomial dividing every term of a polynomial. It reverses distribution: $ab+ac=a(b+c)$.
GCF factoring is the first step of almost every polynomial factorization and supports simplification, equation solving, and structural interpretation.
Intuition and core definition
Greatest common factor factoring extracts the largest monomial dividing every term of a polynomial. It reverses distribution: $ab+ac=a(b+c)$. The numerical part is the GCF of coefficients, and each variable uses the smallest exponent present in every term.
Notation, language, and conditions
For coefficients with mixed signs, the GCF is normally positive, but factoring out a negative can make the leading term inside positive. A variable absent from one term has exponent zero, so it is not common. Factoring $0$ or the zero polynomial requires special care; the standard method assumes nonzero terms.
Why this idea matters
Extracting the greatest common factor rewrites a polynomial as a product while leaving the remaining polynomial primitive over the chosen coefficient set.
A dependable method
- Find the numerical GCF of all coefficients.
- For each variable present in every term, choose the minimum exponent.
- Choose the sign of the common factor deliberately.
- Divide each original term by the GCF to build the bracket.
- Distribute the factor back and confirm every term.
