Factoring Polynomials – Greatest Common Monomial Factor

Key Concepts

• Factoring = rewriting a polynomial as a product of simpler factors.

• Greatest Common Monomial Factor (GCMF): highest-degree monomial dividing every term with no remainder.

• A polynomial is composite if it factors; prime if no non-trivial factors exist.

Steps to Find the GCMF

• Separate numerical coefficients and variables.

• Numerical part: take the greatest common factor (GCF) of the coefficients.

• Variable part: for each common variable, use the lowest exponent present in all terms.

• Product of numerical and variable GCFs = GCMF.
  Example: $3x^2y - 12y \rightarrow \text{GCMF}=3y$.

Factoring With the GCMF

1. Determine the GCMF.

2. Divide each term by the GCMF.

3. Rewrite: $\text{Polynomial}=\text{GCMF}\times(\text{quotient})$.

4. Check by multiplying factors to recover the original expression.
   Example: $3x^2y - 12y = 3y(xy-4)$.

Practical Applications

• Engineering: simplify force, stress, or design polynomials.

• Economics: streamline cost & revenue models.

• Science: balance equations, model rates & motions.

Useful Patterns / Quick Recall

• Numerical GCF often equals the smallest coefficient when all coefficients share that divisor.

• Missing variable in any term ⇒ variable not part of GCMF.

• Area problems: remaining area = whole − part, then factor.
  Circle minus square (radius $r=s$): $A=s^2(\pi-1)$.
  Square minus circle (diameter $=s$): $A=s^2\bigl(1-\frac{\pi}{4}\bigr)$.

Common Exercise Types

• Basic factoring: e.g.
  $45a^4+36a^3 = 9a^3(5a+4)$.

• Multivariable: e.g.
  $24x^3y^2+18x^3y^2-30xy^3 = 6xy^2(4x^2+3x^2-5y)$.

• Proof of compositeness: show non-trivial GCMF exists (e.g., $75m^5+15m^3+45m^2=15m^2(5m^3+m+3)$).

Checklist for Factoring Success

• [ ] List coefficient factors quickly.

• [ ] Scan variables for lowest exponents.

• [ ] Pull out GCMF first before other techniques.

• [ ] Always verify by distributing.