How to Factor Out the Greatest Common Factor (GCF) from Polynomials

📚 Quick Study Guide

🔍 Greatest Common Factor (GCF): The largest factor that divides two or more numbers or terms. 🔢 Finding the GCF of Numbers: List the factors of each number and identify the largest one they share. 🧮 Finding the GCF of Variables: Identify the lowest power of each variable present in all terms. 💡 Factoring out the GCF: Divide each term in the polynomial by the GCF and write the polynomial as GCF(remaining terms). 📝 Example: To factor $4x^2 + 6x$, the GCF is $2x$. Factoring it out gives $2x(2x + 3)$. 🧪 Checking Your Work: Distribute the GCF back into the parentheses to ensure you get the original polynomial.

Practice Quiz

1. What is the GCF of $12x^3$ and $18x^2$?
   1. $6x$
   2. $6x^2$
   3. $36x^3$
   4. $3x$
2. Factor out the GCF from the polynomial $5y^4 - 10y^2$.
   1. $5y^2(y^2 - 2)$
   2. $5y^2(y^2 - 10)$
   3. $5(y^4 - 2y^2)$
   4. $y^2(5y^2 - 10)$
3. What is the GCF of $8a^2b$ and $12ab^2$?
   1. $4ab$
   2. $2ab$
   3. $4a^2b^2$
   4. $2a^2b^2$
4. Factor out the GCF from the polynomial $9m^3n + 15mn^2$.
   1. $3mn(3m^2 + 5n)$
   2. $3m(3m^2n + 5n^2)$
   3. $3n(3m^3 + 5mn)$
   4. $mn(9m^2 + 15n)$
5. What is the GCF of $25p^5$ and $15p^3$?
   1. $5p^3$
   2. $5p^2$
   3. $5p^5$
   4. $15p^3$
6. Factor out the GCF from the polynomial $7z^6 - 14z^3 + 21z$.
   1. $7z(z^5 - 2z^2 + 3)$
   2. $7(z^6 - 2z^3 + 3z)$
   3. $z(7z^5 - 14z^2 + 21)$
   4. $7z(z^6 - 14z^3 + 21z)$
7. What is the GCF of $3x^2y$, $6xy^2$, and $9x^3y^3$?
   1. $3xy$
   2. $3x^2y^2$
   3. $9x^3y^3$
   4. $x^2y^2$