Examples of factoring polynomials using GCF

📚 Quick Study Guide: Factoring Polynomials with GCF

Factoring polynomials using the Greatest Common Factor (GCF) is a fundamental skill in algebra. It's essentially the reverse of distribution, allowing us to simplify expressions and solve equations.

• 💡 What is Factoring? It's the process of breaking down a polynomial into a product of simpler polynomials (its factors). Think of it like finding the prime factors of a number, but for algebraic expressions!
• 🔍 What is the GCF? The Greatest Common Factor (GCF) of a set of terms is the largest monomial that divides evenly into each term of the polynomial. This includes both numerical coefficients and variable parts.
• 🔢 Steps to Factor using GCF:
• 🎯 Step 1: Identify the GCF of Coefficients. Find the largest number that divides into all numerical coefficients.
• 🌱 Step 2: Identify the GCF of Variables. For each common variable, take the one with the lowest exponent present in all terms. If a variable isn't in all terms, it's not part of the GCF for variables.
• 🔗 Step 3: Combine to Form the Overall GCF. Multiply the GCF of the coefficients by the GCF of the variables.
• ➗ Step 4: Divide Each Term by the GCF. Perform polynomial division for each term of the original polynomial by the GCF you found.
• 📝 Step 5: Write in Factored Form. The factored form is the $GCF(quotient)$, where 'quotient' is the polynomial you get after dividing each term.
• 🌟 Example: Let's factor $6x^2 + 9x$.
• 🧐 Coefficients: 6 and 9. Their GCF is 3.
• 📏 Variables: $x^2$ and $x$. The lowest exponent for $x$ is 1 (from $x^1$). So, the GCF of variables is $x$.
• 🔑 Overall GCF: $3 \times x = 3x$.
• ⚙️ Divide each term: $(6x^2)/(3x) = 2x$ and $(9x)/(3x) = 3$.
• ✅ Factored form: $3x(2x+3)$.

🧠 Practice Quiz: Factoring Polynomials

Test your knowledge with these multiple-choice questions!

1. Which of the following is the GCF of the polynomial $12x^3 - 18x^2$?
   1. $3x^2$
   2. $6x$
   3. $6x^2$
   4. $12x^3$
2. Factor the polynomial $15a^2b - 20ab^2$.
   1. $5ab(3a - 4b)$
   2. $5ab(3a - 4ab)$
   3. $5ab(3a - 4)$
   4. $5ab(3a^2 - 4b)$
3. What is the completely factored form of $-8y^3 + 4y^2 - 12y$?
   1. $-4y(2y^2 - y + 3)$
   2. $4y(-2y^2 + y - 3)$
   3. $4y(2y^2 - y + 3)$
   4. $-4y(2y^2 + y - 3)$
4. Factor $7m^2n^3 + 21m^3n^2 - 14m^2n^2$.
   1. $7m^2n^2(n + 3m - 2)$
   2. $7m^2n^2(n^3 + 3m^3 - 2n^2)$
   3. $7m^2n^2(n + 3m - 2mn)$
   4. $7m^2n(n^2 + 3mn - 2n)$
5. Which expression represents the factored form of $10x^4 - 25x^3 + 5x^2$?
   1. $5x^2(2x^2 - 5x)$
   2. $5x^2(2x^2 - 5x + 1)$
   3. $5x^2(2x^2 - 5x + x)$
   4. $5x(2x^3 - 5x^2 + x)$
6. If the GCF of $Ax^3 + Bx^2$ is $3x^2$, which of the following could be the values for A and B?
   1. A=6, B=9
   2. A=4, B=7
   3. A=9, B=4
   4. A=12, B=8
7. A rectangular garden has an area given by the polynomial $9x^2y + 15xy^2$ square meters. If the width of the garden is $3xy$ meters, what is the length?
   1. $3x + 5y$ meters
   2. $3xy + 5y$ meters
   3. $6x + 12y$ meters
   4. $3x^2y + 5xy^2$ meters