Factoring by Grouping

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Just like the name implies, factoring by grouping means that you will group terms with common factors before factoring. Among various factoring techniques, factoring by grouping is particularly useful for trinomials and higher-degree polynomials where other methods might be cumbersome. This lesson delves deep into factoring by grouping, providing clear explanations and multiple examples to ensure a thorough understanding.

What is Factoring by Grouping?

Factoring by grouping involves organizing the terms of a polynomial into groups that share a common factor. Once grouped, each set of terms is factored individually, and if done correctly, a common binomial factor emerges, allowing the entire polynomial to be factored completely.

More Examples Explaining Factoring by Grouping

Let's explore several examples to illustrate the factoring by grouping process.

Example 1: Factoring a Simple Quadratic

Problem: Factor x² + 5x + 6

Step 1: Analyze the Polynomial

• The polynomial has three terms, making it a trinomial.
• To use factoring by grouping, we need to express it as a four-term polynomial.

Step 2: Expand the Middle Term

• Find two numbers that multiply to 1 × 6 = 6 and add up to 5.
• 5x = 3x + 2x

Step 3: Rewrite the Polynomial

x² + 5x + 6 = x² + 3x + 2x + 6

Step 4: Group Terms

• Group the first two terms and the last two terms: (x² + 3x) + (2x + 6)

Step 5: Factor Each Group

• Factor out the GCF from each group: x(x + 3) + 2(x + 3)

Step 6: Factor Out the Common Binomial

• Both groups contain (x + 3): (x + 3)(x + 2)

Conclusion: x² + 5x + 6 = (x + 3)(x + 2)

Example 2: A Slightly More Complex Quadratic

Problem: Factor x² − 4x − 12

Step 1: Analyze the Polynomial

• The polynomial is a trinomial with coefficients and/or minus sign(s) that complicate simple factoring.

Step 2: Expand the Middle Term

• Find two numbers that multiply to 1 × (−12) = − 12 and add up to -4.
  
  
• Initial attempts:
  −2x + (−2x)→ Sum: -4, Product: 4 (Incorrect)
  
  −3x + (−x) → Sum: -4, Product: 3x (Incorrect)
  
  
• Correct Combination: −6x + 2x → Sum: -4x, Product: -12

Step 3: Rewrite the Polynomial: x² − 4x − 12 = x² − 6x + 2x − 12

Step 4: Group Terms: (x² − 6x) + ( 2x − 12)

Step 5: Factor Each Group: x(x − 6) + 2(x − 6)

Step 6: Factor Out the Common Binomial: (x − 6)(x + 2)

Conclusion: x² − 4x − 12 = (x − 6)(x + 2)

Example 3: Factoring with Coefficients

Problem: Factor 3y² + 14y + 8

Step 1: Analyze the Polynomial

• The trinomial has a leading coefficient other than 1, which requires careful grouping.

Step 2: Expand the Middle Term

• Multiply the leading coefficient by the constant: 3 × 8 = 24. 
• Find two numbers that multiply to 24 and add to 14: 12 and 2.
• 14y = 12y + 2y

Step 3: Rewrite the Polynomial: 3y² + 14y + 8 = 3y² + 12y + 2y + 8

Step 4: Group Terms: (3y² + 12y) + (2y + 8)

Step 5: Factor Each Group: 3y(y + 4) +2(y + 4)

Step 6: Factor Out the Common Binomial: (y + 4)(3y + 2)

Conclusion: 3y² + 14y + 8 = (y + 4)(3y + 2)

Example 4: A More Complex Quadratic

Problem: Factor 11x² − 41x − 12

Step 1: Analyze the Polynomial

• The trinomial has large coefficients, making simple factoring challenging.

Step 2: Multiply the Leading Coefficient and Constant

• 11 × (−12) = −132

Step 3: Find Two Numbers That Multiply to -132 and Add to -41

• Potential pairs:
   .....
   .....
  -44 and 3→ Sum: −41
  -46 and 5
  -45 and 4
  -40 + -1
  -39 + -2
  -38 + -3
  -37 + -4
  .....
  .....
• Correct Pair: −44 and 3 since this is the one that is equal to -132 when multiplied

Step 4: Rewrite the Polynomial: 11x² − 41x − 12 = 11x² − 44x + 3x − 12

Step 5: Group Terms: (11x² − 44x) + (3x − 12)

Step 6: Factor Each Group: 11x(x − 4)+ 3(x − 4)

Step 7: Factor Out the Common Binomial: (x − 4)(11x + 3)

Conclusion: 11x² − 41x − 12 = (x − 4)(11x + 3)

Common Pitfalls and Tips when Factoring by Grouping

Incorrect Grouping: Not all groupings will lead to a common binomial factor. Ensure that the numbers chosen for expanding the middle term are correct.

Multiple Groupings: Sometimes, multiple groupings are possible. Test each to find the one that results in a common binomial factor.

Leading Coefficient Greater Than 1: When the leading coefficient isn't 1, multiply it by the constant term to find suitable factors for the middle term.

Verification: Always multiply the factored binomials to verify the correctness of the factoring.

Advanced Example: Applying the Technique

Problem: Factor 12x³ + 8x² − 18x − 12

Step 1: Analyze the Polynomial

• This is a four-term polynomial, making it suitable for factoring by grouping.

Step 2: Group Terms: (12x³ + 8x²)+(−18x − 12)

Step 3: Factor Each Group:

• First group: 12x³ + 8x² = 4x²(3x + 2)
• Second group: −18x − 12 = −6(3x + 2)

Step 4: Factor Out the Common Binomial:

4x²(3x + 2) − 6(3x + 2) = (3x + 2)(4x² − 6)

Step 5: Factor Further if Possible

• 4x² − 6 = 2(2x² − 3)

Final Factored Form: 12x³ + 8x² − 18x − 12 = 2(3x + 2)(2x² − 3)