factoring polynomials examples

📚 Quick Study Guide

🔍 Factoring is the process of breaking down a polynomial into simpler terms (its factors) that, when multiplied together, give you the original polynomial.

💡 Common Factoring Techniques:
• 🍎 Greatest Common Factor (GCF): Find the largest factor common to all terms and factor it out.
• 🧮 Difference of Squares: Use the formula $a^2 - b^2 = (a + b)(a - b)$.
• 📐 Perfect Square Trinomials: Use the formulas $a^2 + 2ab + b^2 = (a + b)^2$ or $a^2 - 2ab + b^2 = (a - b)^2$.
• ➕ Factoring by Grouping: Group terms and factor out common factors from each group.
• ✂️ Trial and Error: For quadratic trinomials of the form $ax^2 + bx + c$, find two numbers that multiply to $ac$ and add up to $b$.

Practice Quiz

1. What is the greatest common factor (GCF) of $12x^3 + 18x^2 - 24x$?
   1. $6x$
   2. $12x$
   3. $6x^2$
   4. $12x^2$
2. Factor the expression: $x^2 - 9$
   1. $(x - 3)(x - 3)$
   2. $(x + 3)(x + 3)$
   3. $(x + 3)(x - 3)$
   4. $(x - 9)(x + 1)$
3. Factor the expression: $4x^2 + 12x + 9$
   1. $(2x + 3)(2x - 3)$
   2. $(2x + 3)^2$
   3. $(2x - 3)^2$
   4. $(4x + 9)(x + 1)$
4. Factor the expression: $x^2 + 5x + 6$
   1. $(x + 1)(x + 6)$
   2. $(x + 2)(x + 3)$
   3. $(x + 5)(x + 1)$
   4. $(x - 2)(x - 3)$
5. Factor the expression: $2x^2 - 8$
   1. $2(x - 2)(x + 2)$
   2. $2(x - 4)(x + 4)$
   3. $(2x - 4)(x + 2)$
   4. $(x - 2)(2x + 4)$
6. Factor by grouping: $x^3 + 2x^2 + 3x + 6$
   1. $(x^2 + 3)(x + 2)$
   2. $(x^2 + 2)(x + 3)$
   3. $(x^2 + 6)(x + 1)$
   4. $(x^2 + 1)(x + 6)$
7. Factor completely: $3x^2 - 12x + 12$
   1. $3(x - 2)^2$
   2. $3(x + 2)^2$
   3. $(3x - 6)^2$
   4. $3(x^2 - 4x + 4)$