Real-World Examples of the Greatest Common Factor in Algebra

📚 Quick Study Guide

• 🔢 The Greatest Common Factor (GCF) is the largest factor that divides two or more numbers (or algebraic terms) without any remainder.
• ➗ In algebra, GCF is used to simplify expressions by factoring out the largest common term.
• ✏️ Factoring out the GCF involves identifying the largest number and the highest power of variables common to all terms.
• ➕ When distributing back the GCF, ensure it multiplies correctly with each term inside the parentheses to yield the original expression.
• 💡 Real-world applications include simplifying ratios, dividing resources, and solving equations more efficiently.

🧪 Practice Quiz

1. Question 1: What is the GCF of $12x^2$ and $18x$?
   1. $6x^2$
   2. $36x^2$
   3. $6x$
   4. $3x$
2. Question 2: Factor out the GCF from the expression $25a^3b^2 + 15a^2b^3$.
   1. $5a^2b^2(5a + 3b)$
   2. $5a^3b^3(5 + 3)$
   3. $5ab(5a^2b + 3ab^2)$
   4. $5a^2b(5a + 3b^2)$
3. Question 3: A rectangular garden has a length of $16y^3$ meters and a width of $24y^2$ meters. What is the largest monomial that can represent both dimensions?
   1. $4y$
   2. $8y^2$
   3. $4y^2$
   4. $8y^3$
4. Question 4: Simplify the expression $\frac{36p^4q^2}{48p^2q^3}$ by factoring out the GCF.
   1. $\frac{3p^2}{4q}$
   2. $\frac{3p^2q}{4}$
   3. $\frac{3p^4}{4q}$
   4. $\frac{3p^2}{4q^2}$
5. Question 5: Find the GCF of $42m^5n$ and $28m^3n^2$.
   1. $14m^3n$
   2. $7m^3n$
   3. $14m^5n^2$
   4. $28m^3n$
6. Question 6: Factor out the GCF from $14c^4d^3 - 21c^2d^5$.
   1. $7c^2d^3(2c^2 - 3d^2)$
   2. $7c^4d^5(2 - 3)$
   3. $7c^2d^2(2c^2d - 3d^3)$
   4. $7c^2d^3(2c^2 - 3d^2)$
7. Question 7: What is the GCF of $75x^4y^6$ and $45x^3y^4$?
   1. $15x^3y^4$
   2. $5x^3y^4$
   3. $15x^4y^6$
   4. $45x^3y^4$