GCF: 8
Teacher Answer Key • Factor Foundations Turn to Page 3 for Numbers with Variables → Page 2 of 4
Teacher Key Monomial Factors & Complete Prime Expansion
Page 3 Key
Part 2 • Algebraic Monomials
Prime factor decomposition and fully expanded algebraic terms
Variables
Grading Note • Monomial Factorization:
Full credit requires both the coefficient factored into prime numbers and the variable powers expanded into linear repeated factors. Any subset of these multiplied factors forms a valid monomial factor.
Problem 4: Factor \(18a^3\) completely. Show all 3 steps. Target: \(18a^3\)
Step 1: Prime factor coefficient 18:
\(18 = 2 \times 9 = \mathbf{2 \cdot 3 \cdot 3}\)
(Factor tree: 18 → 2 and 9; 9 → 3 and 3)
Step 2: Expand variable \(a^3\):
\(a^3 = \mathbf{a \cdot a \cdot a}\)
Step 3: Complete expansion & sample factors:
\(\mathbf{2 \cdot 3 \cdot 3 \cdot a \cdot a \cdot a}\) • Factors: \(3a, 6a^2, 9a, 2a^3\)
Problem 5: Factor \(24x^2y\) completely. Show all 3 steps. Target: \(24x^2y\)
Step 1: Prime factor coefficient 24:
\(24 = 4 \times 6 = \mathbf{2 \cdot 2 \cdot 2 \cdot 3}\)
(Factor tree: 24 → 4 and 6; 4 → 2,2; 6 → 2,3)
Step 2: Expand variables \(x^2y\):
\(x^2y = \mathbf{x \cdot x \cdot y}\)
Step 3: Complete expansion & sample factors:
\(\mathbf{2 \cdot 2 \cdot 2 \cdot 3 \cdot x \cdot x \cdot y}\) • Factors: \(4x, 6xy, 12x^2\)
Teacher Answer Key • Factor Foundations Turn to Page 4 for Factor Verification & GCF → Page 3 of 4
Teacher Key Monomial Factor Testing & GCF Calculations
Page 4 Key
Detailed division simplification, exponent rules, and greatest common factors
Monomial Divisibility Criterion:
When dividing \(\frac{\text{Dividend}}{\text{Divisor}}\), all numerical coefficients must divide evenly, and each variable's power in the dividend must be \(\ge\) the divisor's power so no variables remain in the denominator.
Problem 6: Is \(4ab\) a factor? YES
\(\frac{20a^2b^3}{4ab} = \left(\frac{20}{4}\right)\left(\frac{a^2}{a}\right)\left(\frac{b^3}{b}\right)\)
\(= \mathbf{5 \cdot a^{2-1} \cdot b^{3-1}} = \mathbf{5ab^2}\)
• No remainder • Whole polynomial
Conclusion:
\(4ab\) IS a factor.
Other factor: \(5ab^2\)
Problem 7: Is \(6x^2\) a factor? NO
\(\frac{18xy^2}{6x^2} = \left(\frac{18}{6}\right)\left(\frac{x}{x^2}\right)\left(y^2\right)\)
\(= \mathbf{\frac{3y^2}{x}}\) (\(x\) remains in denominator!)
• Target only has \(x^1\), need \(x^2\)
Conclusion:
\(6x^2\) IS NOT a factor.
Reason: Insufficient powers of \(x\) in target.
Problem 8: GCF of \(15m^3\) and \(25m^2\) \(\text{GCF} = 5m^2\)
Step 1: Fully expand \(15m^3\):
\(15m^3 = 3 \cdot \mathbf{5 \cdot m \cdot m} \cdot m\)
Step 2: Fully expand \(25m^2\):
\(25m^2 = 5 \cdot \mathbf{5 \cdot m \cdot m}\)
Step 3: Shared common factors: \(5 \cdot m \cdot m\) • Final GCF: \(\text{GCF} = 5m^2\)
Teacher Answer Key • Factor Foundations All 8 Problems Complete • Answer Key Page 4 of 4
