Factoring Trinomials by Grouping

Yesterday, we refreshed our skills on factoring out the Greatest Common Factor (GCF). Today, we're leveling up! We'll explore how to factor trinomials that can't be factored using just the GCF. This method, called "factoring by grouping," might seem a bit longer at first, but it provides a consistent approach that works for a wider range of problems. So, buckle up and let's dive in!

Understanding the Process

Factoring by grouping is particularly useful when you have a trinomial in the form $ax^2 + bx + c$, where $a$ is not equal to 1. Here's a breakdown of the steps:

  1. Multiply $a$ and $c$: Calculate the product of the coefficient of the $x^2$ term ($a$) and the constant term ($c$). Let's call this product $ac$.
  2. Find Factors of $ac$: Find two numbers that multiply to $ac$ and add up to $b$ (the coefficient of the $x$ term). These two numbers will help us rewrite the middle term.
  3. Rewrite the Trinomial: Replace the middle term ($bx$) with the two factors you found in the previous step. This will give you a four-term expression.
  4. Factor by Grouping: Group the first two terms and the last two terms. Factor out the GCF from each group. Ideally, you'll have a common binomial factor.
  5. Final Factorization: Factor out the common binomial factor. The result will be the factored form of the original trinomial.

Example 1

Let's factor the trinomial $4x^2 + 4x - 3$:

  • Multiply $a$ and $c$: $4 * -3 = -12$
  • Find factors of -12 that add up to 4: The factors are 6 and -2 (because $6 * -2 = -12$ and $6 + (-2) = 4$)
  • Rewrite the trinomial: $4x^2 + 6x - 2x - 3$
  • Factor by grouping: $2x(2x + 3) - 1(2x + 3)$
  • Final factorization: $(2x + 3)(2x - 1)$

Therefore, $4x^2 + 4x - 3 = (2x + 3)(2x - 1)$

Example 2

Let's factor the trinomial $2x^2 + x - 10$:

  • Multiply $a$ and $c$: $2 * -10 = -20$
  • Find factors of -20 that add up to 1: The factors are 5 and -4 (because $5 * -4 = -20$ and $5 + (-4) = 1$)
  • Rewrite the trinomial: $2x^2 + 5x - 4x - 10$
  • Factor by grouping: $x(2x + 5) - 2(2x + 5)$
  • Final factorization: $(2x + 5)(x - 2)$

Therefore, $2x^2 + x - 10 = (2x + 5)(x - 2)$

Practice Problems

Ready to test your skills? Try factoring these trinomials using the grouping method:

  • $3x^2 + 13x + 4$
  • $9x^2 + 12x + 4$
  • $3x^2 + x - 4$

Homework

Complete problems #37-45 on page 82 of your textbook to solidify your understanding.

Discussion Question of the Day

Everyone always wants to know the shortest way to do a problem because they just want to get it done. What would you rather have: "tricks that make some problems real easy but you have to remember when they work and when they don't" or "a process that always works but might take a bit longer for some problems"? Pick your option and explain why you chose it. There is no wrong answer, but it will give me information on how you want to learn.