Factoring: 12m²n + 24m³n² - 36m⁴n³ + 48m⁵n⁴

Hey guys! Let's dive into factoring this polynomial expression. Factoring is like reverse multiplication, where we break down an expression into its simpler multiplicative components. It's a fundamental skill in algebra, and it's super useful for simplifying expressions, solving equations, and even in more advanced math topics. So, let's get started!

Identifying the Greatest Common Factor (GCF)

When we're faced with an expression like 12m²n + 24m³n² - 36m⁴n³ + 48m⁵n⁴, the first step is always to find the greatest common factor (GCF). The GCF is the largest term that divides evenly into all the terms in the expression. Think of it as the biggest piece we can pull out of every term.

So, how do we find the GCF? Well, we need to look at both the coefficients (the numbers) and the variables (the letters) in each term. Let's break it down:

• Coefficients: We have 12, 24, -36, and 48. What's the biggest number that divides into all of these? It's 12! So, 12 is part of our GCF.
• Variables (m): We have m², m³, m⁴, and m⁵. The smallest exponent of 'm' is 2, so m² is part of our GCF. We take the lowest power of the common variable.
• Variables (n): We have n, n², n³, and n⁴. The smallest exponent of 'n' is 1 (just 'n'), so 'n' is part of our GCF. Same as with 'm', we take the lowest power.

Putting it all together, the greatest common factor (GCF) is 12m²n. This is the key to simplifying our expression. Identifying the GCF is a crucial step because it allows us to reduce the complexity of the expression and make it easier to work with. Once we've found the GCF, we can move on to the next step: factoring it out.

Factoring Out the GCF

Now that we've identified the GCF as 12m²n, the next step is to factor it out of the original expression. This is where we essentially divide each term in the expression by the GCF and write the result in a factored form. It might sound a bit complicated, but it's actually quite straightforward once you get the hang of it.

Here's how we do it:

1. Write down the GCF: 12m²n
2. Open a set of parentheses: 12m²n(
3. Divide each term in the original expression by the GCF and write the result inside the parentheses:
   • (12m²n) / (12m²n) = 1
   • (24m³n²) / (12m²n) = 2mn
   • (-36m⁴n³) / (12m²n) = -3m²n²
   • (48m⁵n⁴) / (12m²n) = 4m³n³
4. Write the results inside the parentheses, separated by the appropriate signs: 1 + 2mn - 3m²n² + 4m³n³
5. Close the parentheses: 12m²n(1 + 2mn - 3m²n² + 4m³n³)

So, after factoring out the GCF, our expression looks like this: 12m²n(1 + 2mn - 3m²n² + 4m³n³). We've successfully factored out the greatest common factor, which means we've taken the largest term that divides evenly into all parts of the expression and placed it outside the parentheses. The expression inside the parentheses represents what's left after we've divided each original term by the GCF. This is a significant step in simplifying the original expression and making it easier to analyze or manipulate further.

Checking Your Work

Alright, we've factored out the GCF, but how can we be sure we did it right? It's always a good idea to check your work, especially in math. The easiest way to check our factoring is to distribute the GCF back into the parentheses. This is basically the reverse of factoring, and it should give us back our original expression.

Let's do it:

We have 12m²n(1 + 2mn - 3m²n² + 4m³n³).

Now, we'll distribute the 12m²n to each term inside the parentheses:

• 12m²n * 1 = 12m²n
• 12m²n * 2mn = 24m³n²
• 12m²n * (-3m²n²) = -36m⁴n³
• 12m²n * 4m³n³ = 48m⁵n⁴

Now, let's put these terms back together: 12m²n + 24m³n² - 36m⁴n³ + 48m⁵n⁴

Hey, look at that! It's exactly the same as our original expression. This means we factored out the GCF correctly. Checking your work is a critical step in any math problem. It helps you catch mistakes and ensures that your answer is accurate. By distributing the GCF back into the parentheses, we've confirmed that our factored expression is equivalent to the original, giving us confidence in our solution.

Further Simplification (If Possible)

Okay, we've factored out the GCF and checked our work. Awesome! Now, let's think about whether we can simplify further. Sometimes, the expression inside the parentheses can be factored even more. This usually involves looking for patterns like difference of squares, perfect square trinomials, or other factoring techniques.

In our case, we have the expression (1 + 2mn - 3m²n² + 4m³n³) inside the parentheses. At first glance, it might not be obvious if this can be factored further. Factoring polynomials with more than three terms can sometimes be tricky, and there isn't always a straightforward method.

To determine if further factoring is possible, we would typically look for:

• Common Factors within the Parentheses: Are there any factors common to all the terms inside the parentheses? In this case, no.
• Special Patterns: Does the expression fit a pattern like difference of squares (a² - b²), perfect square trinomial (a² + 2ab + b²), or sum/difference of cubes (a³ ± b³)? This doesn't seem to fit any of these common patterns.
• Grouping: Can we group terms in a way that reveals a common factor? This might be a possibility, but it's not immediately clear.

In this specific case, after careful consideration, it doesn't appear that the expression inside the parentheses can be factored further using elementary factoring techniques. It's possible that there might be more advanced methods or techniques that could be applied, but for our purposes, we can consider this expression to be in its simplest factored form.

So, our final factored expression is 12m²n(1 + 2mn - 3m²n² + 4m³n³). We've taken out the greatest common factor, and we've examined the remaining expression to see if there are any more straightforward factoring steps we can take. Since we can't simplify it further with basic techniques, we can confidently say that we've factored the original expression as much as possible.

Final Factored Form

Alright, we've journeyed through the entire factoring process, and it's time to present our final factored form. After identifying the GCF, factoring it out, checking our work, and considering further simplification, we've arrived at our solution.

So, the factored form of the expression 12m²n + 24m³n² - 36m⁴n³ + 48m⁵n⁴ is:

12m²n(1 + 2mn - 3m²n² + 4m³n³)

This is it! We've successfully broken down the original expression into a product of simpler factors. The GCF, 12m²n, is factored out, and the expression inside the parentheses, (1 + 2mn - 3m²n² + 4m³n³), represents the remaining terms after the GCF has been removed. We've also determined that the expression inside the parentheses cannot be factored further using basic factoring techniques, so we can be confident that this is the most simplified factored form.

Factoring is a powerful tool in algebra, and mastering it allows you to simplify complex expressions, solve equations, and gain a deeper understanding of mathematical relationships. By following the steps we've outlined – identifying the GCF, factoring it out, checking your work, and considering further simplification – you can tackle a wide range of factoring problems with confidence.

Great job, guys! You've successfully factored a polynomial expression. Keep practicing, and you'll become a factoring pro in no time!