5 Ways to Factor by Grouping: Worksheet Answers

Introduction to Factoring by Grouping

Factoring by grouping is a method used in algebra to factor a polynomial with four terms into two groups. This technique is particularly useful when you have a quadratic or cubic polynomial that does not obviously fit into simpler factoring methods like difference of squares or common factoring. The goal of factoring by grouping is to identify common factors within smaller sets of terms and use these to factor the entire expression.

Step-by-Step Guide to Factoring by Grouping

Before delving into examples and answers, here's a general outline for factoring by grouping:

• Identify groups: Divide the polynomial into two or more groups, typically with two terms each.
• Factor out common terms: Look for the greatest common factor (GCF) in each group.
• Distribute the common factor: Factor out the GCF from each group, creating a new expression.
• Combine: If there’s a common binomial factor between the two new terms, factor this out to complete the factorization.

Worksheet Problems and Answers

Problem 1

Factor the polynomial: 2x^3 - 6x^2 + 3x - 9.

• Step 1: Group the first two and last two terms:
  • ($2x^3 - 6x^2$) + ($3x - 9$)
• Step 2: Factor out the GCF from each group:
  • $2x^2(x - 3)$ + $3(x - 3)$
• Step 3: Distribute the common factor $(x - 3)$:
  • $x^3(x - 3) + 3(x - 3) = (2x^2 + 3)(x - 3)$

📚 Note: Factoring by grouping can sometimes lead to a final expression that still needs further factoring, especially if the groups are not directly factorable or if there is no common binomial factor.

Problem 2

Factor the polynomial: ax + bx - ay - by.

• Step 1: Group the terms:
  • $(ax + bx) + (-ay - by)$
• Step 2: Factor out the GCF from each group:
  • $x(a + b) - y(a + b)$
• Step 3: Distribute the common binomial factor:
  • $(x - y)(a + b)$

Problem 3

Factor the polynomial: x^2 + 4x - 3x - 12.

• Step 1: Group the terms:
  • $(x^2 + 4x) + (-3x - 12)$
• Step 2: Factor out the GCF from each group:
  • $x(x + 4) - 3(x + 4)$
• Step 3: Distribute the common factor $(x + 4)$:
  • $(x + 4)(x - 3)$

Problem 4

Factor the polynomial: x^3 + 3x^2 - 2x - 6.

• Step 1: Group the terms:
  • $(x^3 + 3x^2) + (-2x - 6)$
• Step 2: Factor out the GCF from each group:
  • $x^2(x + 3) - 2(x + 3)$
• Step 3: Distribute the common factor $(x + 3)$:
  • $(x^2 - 2)(x + 3)$

Problem 5

Factor the polynomial: 3x^2 - 3x + 12x - 12.

• Step 1: Group the terms:
  • $(3x^2 - 3x) + (12x - 12)$
• Step 2: Factor out the GCF from each group:
  • $3x(x - 1) + 12(x - 1)$
• Step 3: Distribute the common factor $(x - 1)$:
  • $(3x + 12)(x - 1)$

📚 Note: Sometimes the terms can be grouped differently to find the factorization. Flexibility in grouping can be key to solving some factoring problems.

When you are able to identify common factors and group terms effectively, factoring by grouping becomes a powerful tool in algebra. It allows for the simplification of complex expressions, making them easier to solve or work with in further mathematical operations.

In closing, factoring by grouping provides a systematic approach to breaking down polynomials into manageable parts. This method not only aids in solving equations but also enhances your understanding of the algebraic structures involved. Remember that not all polynomials are easily factored by grouping; however, when applicable, this technique can reveal deeper patterns in your algebraic expressions.