Factor $26 R^3 S + 52 R^5 - 39 R^2 S^4$: What's The Result?

Hey guys, let's dive into the awesome world of algebra and tackle this factoring problem! We've got the expression $26 r^3 s+52 r^5 - 39 r^2 s^4$, and our mission, should we choose to accept it, is to find out what the resulting expression is after we factor it. This isn't just about crunching numbers; it's about understanding the underlying structure of mathematical expressions and how we can simplify them. Factoring is a fundamental skill in mathematics, kind of like learning your scales before you can play a symphony. It allows us to rewrite complex expressions in a simpler, more manageable form, which is super useful for solving equations, simplifying fractions, and understanding the behavior of functions. So, grab your calculators, your notebooks, and let's break this down step-by-step!

Understanding the Expression and the Goal of Factoring

Before we jump into factoring, let's get a good grip on what we're working with. Our expression is $26 r^3 s+52 r^5 - 39 r^2 s^4$. We have three terms, and each term is a product of coefficients and variables raised to certain powers. The variables here are $r$ and $s$. The goal of factoring, in this case, is to find the Greatest Common Factor (GCF) of all the terms and pull it out. Think of it like finding the biggest possible piece you can remove from a puzzle that fits into all the parts. Once we identify this GCF, we divide each term in the original expression by it, and what's left inside the parentheses is the simplified form. It's all about finding common ground, literally!

Step 1: Identify Common Factors in the Coefficients

Our coefficients are 26, 52, and -39. We need to find the greatest common factor among these numbers. Let's list the factors for each:

• 26: 1, 2, 13, 26
• 52: 1, 2, 4, 13, 26, 52
• 39: 1, 3, 13, 39

Looking at these lists, the greatest common factor for 26, 52, and 39 is 13. So, we know that 13 will be part of our factored expression.

Step 2: Identify Common Factors in the Variables (r)

Now, let's look at the variable $r$ in each term:

• Term 1: $r^3$
• Term 2: $r^5$
• Term 3: $r^2$

To find the common factor for $r$, we take the lowest power of $r$ that appears in all terms. In this case, the lowest power is $r^2$. So, $r^2$ is a common factor.

Step 3: Identify Common Factors in the Variables (s)

Finally, let's examine the variable $s$ in each term:

• Term 1: $s^1$ (since $s$ is $s^1$)
• Term 2: No $s$ term (or $s^0$)
• Term 3: $s^4$

Since the second term doesn't have an $s$ at all, there is no common factor of $s$ across all three terms. This is a crucial observation, guys! We can only factor out what's present in every term.

Step 4: Combine the GCFs and Factor

Now we combine the GCFs we found: the coefficient GCF is 13, and the variable GCF is $r^2$. So, our overall GCF for the entire expression is $13 r^2$.

To factor, we pull this GCF out to the front:

$13 r^2 ( ext{something})$

What goes inside the parentheses? We get that by dividing each original term by our GCF ($13 r^2$):

• Term 1: rac{26 r^3 s}{13 r^2} = rac{26}{13} imes rac{r^3}{r^2} imes s = 2 imes r^{(3-2)} imes s = oxed{2rs}
• Term 2: rac{52 r^5}{13 r^2} = rac{52}{13} imes rac{r^5}{r^2} = 4 imes r^{(5-2)} = oxed{4r^3}
• Term 3: rac{-39 r^2 s^4}{13 r^2} = rac{-39}{13} imes rac{r^2}{r^2} imes s^4 = -3 imes r^{(2-2)} imes s^4 = -3 imes r^0 imes s^4 = -3 imes 1 imes s^4 = oxed{-3s^4}

Putting it all together, our factored expression is:

$$13 r^2 (2rs + 4r^3 - 3s^4)$$

This is the simplified form where we've extracted the largest possible common factor. It's like decluttering your mathematical workspace!

Analyzing the Options Provided

Now that we've done the hard work, let's compare our result to the options given:

A. 13old{(2 r^3 s+4 r^5-3 r^2 s^4)}

This option only factored out the coefficient 13, but it missed the common factor of $r^2$. So, this is incorrect.

B. 13 r^2old{(26 r^3 s+52 r^5-39 r^2 s^4)}

This option looks like it just multiplied the original expression by $13r^2$, not factored it. The terms inside the parentheses are the original terms, which means nothing was actually factored out. This is definitely not right.

C. 13 r^2old{(2 r s+4 r^3-3 s^4)}

Boom! This matches our calculated factored expression exactly. We found the GCF to be $13 r^2$, and when we divided the original terms by $13 r^2$, we got $2rs$, $4r^3$, and $-3s^4$. This is our winner, guys!

D. 13 r^2 sold{(2 r+4 r^3-3 s^3)}

This option incorrectly factored out an $s$ from all terms. As we noted in Step 3, there wasn't a common factor of $s$ because the middle term ($52r^5$) didn't have an $s$. So, this is also incorrect.

The Final Answer: A Triumph of Algebraic Decomposition

So, after carefully examining the expression $26 r^3 s+52 r^5 - 39 r^2 s^4$ and applying the principles of factoring, we've arrived at our solution. We identified the greatest common factor (GCF) of the coefficients (13) and the greatest common factor of the variable $r$ (which is $r^2$). We also confirmed that there was no common factor of $s$ across all terms. By dividing each term by the GCF of $13r^2$, we were able to rewrite the expression in its factored form. This process is fundamental in algebra, enabling us to simplify complex expressions and set the stage for solving more intricate problems. The ability to spot common factors and perform this algebraic decomposition is a key skill that will serve you well as you continue your mathematical journey. Remember, every complex problem can often be simplified by breaking it down into its fundamental components, and factoring is a prime example of this powerful technique in action. It's all about finding those common threads that tie the different parts of an expression together and pulling them out to reveal a simpler, more elegant structure. So, when you see an expression like this, don't get intimidated! Just think about finding the GCF, divide, and voilà – you've got a simplified, factored form ready for whatever mathematical adventure comes next. Keep practicing, and you'll become a factoring ninja in no time!

Therefore, the correct resulting expression is C. 13 r^2old{(2 r s+4 r^3-3 s^4)}.