Algebra 1 Test Questions on Multi-Step Polynomial Factoring

📚 Quick Study Guide

• 💡Greatest Common Factor (GCF): Always factor out the GCF first. This simplifies the polynomial and makes subsequent factoring easier.
• ➗Factoring by Grouping: Useful for polynomials with four terms. Group terms, factor out common factors from each group, and then factor out the common binomial.
• 🧮Difference of Squares: Recognize the pattern $a^2 - b^2 = (a + b)(a - b)$.
• ➕Perfect Square Trinomials: Identify the pattern $a^2 + 2ab + b^2 = (a + b)^2$ or $a^2 - 2ab + b^2 = (a - b)^2$.
• 📈Trial and Error: For trinomials of the form $ax^2 + bx + c$, use trial and error to find two binomials that multiply to give the original trinomial.
• 📝Sum/Difference of Cubes: Know the formulas $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ and $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.
• ✔️Always Check: After factoring, always check your answer by multiplying the factors back together to ensure you get the original polynomial.

🧪 Practice Quiz

1. What is the first step in factoring $6x^3 + 15x^2 - 9x$ completely?
   1. A) Factor out $3x$
   2. B) Factor out $6$
   3. C) Factor out $x^2$
   4. D) Factor by grouping
2. Factor completely: $x^2 - 49$
   1. A) $(x - 7)^2$
   2. B) $(x + 7)(x - 7)$
   3. C) $(x + 7)^2$
   4. D) $(x - 49)(x + 1)$
3. Factor completely: $2x^2 + 5x + 2$
   1. A) $(2x + 1)(x + 2)$
   2. B) $(2x + 2)(x + 1)$
   3. C) $(x + 1)(x + 2)$
   4. D) $(2x - 1)(x - 2)$
4. Factor completely: $x^3 + 8$
   1. A) $(x + 2)(x^2 + 2x + 4)$
   2. B) $(x + 2)(x^2 - 2x + 4)$
   3. C) $(x - 2)(x^2 + 2x + 4)$
   4. D) $(x - 2)(x^2 - 2x + 4)$
5. Factor completely: $3x^2 - 12$
   1. A) $3(x - 2)^2$
   2. B) $3(x + 2)(x - 2)$
   3. C) $(3x + 6)(x - 2)$
   4. D) $(x + 2)(3x - 6)$
6. Factor by grouping: $x^3 + 2x^2 + 3x + 6$
   1. A) $(x^2 + 3)(x + 2)$
   2. B) $(x^2 + 2)(x + 3)$
   3. C) $(x^2 + 6)(x + 1)$
   4. D) $(x^2 + 1)(x + 6)$
7. Factor completely: $4x^2 - 20x + 25$
   1. A) $(2x + 5)^2$
   2. B) $(2x - 5)^2$
   3. C) $(4x - 5)(x - 5)$
   4. D) $(2x + 5)(2x - 5)$