Factoring Polynomial X³ + 4x² - 9x - 36 By Grouping
Hey guys! Today, we are diving into the fascinating world of polynomial factorization. We've got a cubic polynomial here, x³ + 4x² - 9x - 36, and our mission is to break it down into its factors using a technique called factoring by grouping. This method is super handy when you have polynomials with four terms, just like this one. So, let's roll up our sleeves and get started!
Understanding the Problem
First, let’s make sure we all understand what we're dealing with. The polynomial we have is x³ + 4x² - 9x - 36. Notice that it has four terms: x³, 4x², -9x, and -36. Factoring is like reverse multiplication; we want to find the expressions that, when multiplied together, give us this original polynomial. Factoring by grouping is a strategy that involves pairing terms and finding common factors within those pairs. It’s a bit like detective work, where we’re piecing together clues to solve the mystery of the factors. This method works particularly well when the polynomial has four terms, because it allows us to create two groups, each of which can be factored separately. By identifying and extracting common factors from each group, we often reveal a common binomial factor that ties the whole expression together. This is the key to successfully factoring the polynomial, and it's what makes factoring by grouping such a powerful technique. So, let's jump right in and see how it works step by step!
Step-by-Step Factoring
Step 1: Group the Terms
The first step in factoring by grouping is, well, to group the terms! We’re going to pair the first two terms and the last two terms together. This gives us:
(x³ + 4x²) + (-9x - 36)
Grouping terms is like forming alliances. We're setting up the polynomial so we can identify common factors within each group more easily. By pairing the terms in this way, we create opportunities to simplify the expression, which is crucial for the next steps in the factoring process. The parentheses act as a visual aid, helping us to keep track of the groups and treat them as separate units for the time being. This strategic grouping is the foundation of factoring by grouping, and it’s what allows us to break down a complex polynomial into more manageable parts. Now that we've grouped our terms, we're ready to move on to the next step: finding the greatest common factor (GCF) in each group. Let's dive into that!
Step 2: Find the Greatest Common Factor (GCF) in Each Group
Now, let's look at each group separately and find the greatest common factor (GCF). In the first group, (x³ + 4x²), the GCF is x². We can factor x² out of both terms:
x²(x + 4)
In the second group, (-9x - 36), the GCF is -9 (yes, we include the negative sign!). Factoring out -9 gives us:
-9(x + 4)
Finding the GCF is like identifying the common thread that runs through each group. It's the largest factor that divides evenly into all terms within the group. In the first group, both x³ and 4x² have x² as a common factor, so we pull that out. In the second group, both -9x and -36 are divisible by -9. It's crucial to include the negative sign here, as it helps us to reveal the common binomial factor in the next step. Once we've identified the GCF in each group, we factor it out, which means dividing each term in the group by the GCF and writing it outside the parentheses. This process simplifies the groups and sets us up perfectly for the final step in factoring by grouping. Let’s see what that is!
Step 3: Factor Out the Common Binomial
Look closely at what we have now:
x²(x + 4) - 9(x + 4)
Notice anything? Both terms have a common factor of (x + 4)! This is the magic of factoring by grouping. We can factor out this common binomial:
(x + 4)(x² - 9)
Spotting the common binomial factor is like finding the missing piece of a puzzle. It's the crucial link that connects the two groups we factored earlier. In this case, both terms share the factor (x + 4). When we factor out this common binomial, we're essentially dividing each term by (x + 4) and writing it outside a new set of parentheses. What remains inside the parentheses are the factors that were multiplied by the common binomial, in this case, x² and -9. This step is where the polynomial starts to take its final factored form. However, we're not quite done yet! There's one more step we need to consider to fully factor the polynomial. Let's check it out in the next section.
Step 4: Factor the Difference of Squares (If Possible)
Now, let’s examine our expression:
(x + 4)(x² - 9)
We can see that (x² - 9) is a difference of squares. Remember the formula: a² - b² = (a + b)(a - b). So, we can factor (x² - 9) as (x + 3)(x - 3).
Therefore, the fully factored form is:
(x + 4)(x + 3)(x - 3)
Recognizing and factoring the difference of squares is like adding the final flourish to a masterpiece. It's the last step in ensuring that our polynomial is completely factored. The term (x² - 9) fits the pattern of a difference of squares because x² is a perfect square and 9 is a perfect square (3²). The difference of squares formula allows us to factor this term into two binomials: (x + 3) and (x - 3). This final factorization gives us the complete set of factors for the original polynomial. It's a neat and tidy solution that breaks down the complex expression into its simplest components. So, with this last step, we've successfully factored the polynomial! Let's take a look at our options now and see which one matches our solution.
Identifying the Correct Factorization
Okay, so we've factored the polynomial x³ + 4x² - 9x - 36 into (x + 4)(x + 3)(x - 3). Now, let's take a look at the options provided and see which one matches our result, keeping in mind that the order of factors doesn't change the product.
Reviewing the Options
We were given the following options:
- (x² + 9)(x + 4)
- (x² - 9)(x - 4)
- (x² + 9)(x - 4)
- (x² - 9)(x + 4)
Matching the Factors
Our factored form is (x + 4)(x + 3)(x - 3). Notice that we factored (x² - 9) into (x + 3)(x - 3). So, we're looking for an option that has (x + 4) and (x² - 9) as factors. Let's analyze each option:
- Option 1: (x² + 9)(x + 4) - This has (x + 4), but (x² + 9) is not the same as (x² - 9), so it’s not the correct answer.
- Option 2: (x² - 9)(x - 4) - This has (x² - 9), but it has (x - 4) instead of (x + 4), so it’s not correct either.
- Option 3: (x² + 9)(x - 4) - Similar to option 1, this has (x² + 9), which is incorrect, and it also has (x - 4) instead of (x + 4).
- Option 4: (x² - 9)(x + 4) - Aha! This has both (x + 4) and (x² - 9), which we know can be further factored into (x + 3)(x - 3). This matches our factored form!
The Correct Answer
Therefore, the correct factorization of x³ + 4x² - 9x - 36 is:
(x² - 9)(x + 4)
This option aligns perfectly with our step-by-step factorization process. We grouped the terms, found the GCF in each group, factored out the common binomial, and then recognized and factored the difference of squares. By carefully following these steps, we arrived at the correct answer. It's like completing a puzzle, where each step fits perfectly into the next, leading us to the final solution. Great job, guys! Let's recap what we've learned and reinforce our understanding of factoring by grouping.
Conclusion
And there you have it! We successfully factored the polynomial x³ + 4x² - 9x - 36 by grouping. Remember, the key steps are: grouping the terms, finding the GCF in each group, factoring out the common binomial, and then checking for any further factoring opportunities (like the difference of squares). Factoring by grouping is a powerful tool in your mathematical toolkit, especially when dealing with polynomials that have four terms. It allows us to break down complex expressions into simpler factors, which can be incredibly useful in solving equations and simplifying expressions. By mastering this technique, you'll be well-equipped to tackle a wide range of polynomial factorization problems. So keep practicing, and you'll become a factoring pro in no time! If you found this helpful, keep an eye out for more math adventures. Until next time, happy factoring!