Factoring: A Step-by-Step Guide To $3s^4 + 30s^3 + 72s^2$

Alright, guys! Let's dive into some algebra and break down how to factor the expression $3s^4 + 30s^3 + 72s^2$. Factoring might seem daunting at first, but with a systematic approach, it becomes a breeze. Trust me, you'll feel like math wizards by the end of this guide! So grab your pencils, and let's get started!

1. Identifying the Greatest Common Factor (GCF)

First things first, when you're faced with an expression like $3s^4 + 30s^3 + 72s^2$, your initial step should always be to identify the greatest common factor (GCF). This is the largest term that divides evenly into all the terms in the expression. In our case, we need to look at both the coefficients (the numbers) and the variable terms (the $s$ terms).

Looking at the coefficients, we have 3, 30, and 72. What's the biggest number that divides into all three of these? Well, 3 divides into 3, 30 (3 x 10), and 72 (3 x 24). So, 3 is our GCF for the coefficients. Next, let's consider the variable terms: $s^4$, $s^3$, and $s^2$. The GCF here is the lowest power of $s$ present in all terms, which is $s^2$. Why $s^2$? Because $s^2$ can divide evenly into $s^4$ (leaving $s^2$), $s^3$ (leaving $s$), and itself (leaving 1).

Therefore, the GCF of the entire expression $3s^4 + 30s^3 + 72s^2$ is $3s^2$. This is the key to simplifying the expression. By identifying and factoring out the GCF, we make the remaining expression much easier to work with. This step is crucial because it reduces the complexity of the problem and sets us up for further factoring if necessary. Always remember to look for the GCF first – it's the golden rule of factoring!

2. Factoring Out the GCF

Now that we've identified our GCF as $3s^2$, the next step is to actually factor it out from the expression $3s^4 + 30s^3 + 72s^2$. Factoring out the GCF means dividing each term in the original expression by the GCF and writing the result in parentheses. So, let's do it step by step:

• Divide $3s^4$ by $3s^2$: $(3s^4) / (3s^2) = s^2$
• Divide $30s^3$ by $3s^2$: $(30s^3) / (3s^2) = 10s$
• Divide $72s^2$ by $3s^2$: $(72s^2) / (3s^2) = 24$

Now, we rewrite the original expression with the GCF factored out: $3s^2(s^2 + 10s + 24)$. What we've done here is essentially reversed the distributive property. If you were to distribute $3s^2$ back into the parentheses, you would get the original expression: $3s^2 * s^2 = 3s^4$, $3s^2 * 10s = 30s^3$, and $3s^2 * 24 = 72s^2$.

This step is super important because it simplifies the expression inside the parentheses, making it easier to factor further. We've essentially pulled out the common part from each term, leaving us with a simpler quadratic expression to deal with. Factoring out the GCF is like taking out a common ingredient from a recipe – it makes the rest of the cooking process much smoother. Always double-check that you've factored out the GCF correctly by redistributing it back into the parentheses to ensure you get the original expression. This way, you can be confident that you're on the right track!

3. Factoring the Quadratic Expression

Alright, now we're at the fun part: factoring the quadratic expression inside the parentheses, which is $s^2 + 10s + 24$. A quadratic expression is in the form of $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In our case, $a = 1$, $b = 10$, and $c = 24$. To factor this, we need to find two numbers that multiply to $c$ (24) and add up to $b$ (10).

Let's think about the factors of 24: 1 and 24, 2 and 12, 3 and 8, 4 and 6. Which of these pairs adds up to 10? Bingo! It's 4 and 6. So, we can rewrite the quadratic expression as $(s + 4)(s + 6)$. To double-check, you can use the FOIL method (First, Outer, Inner, Last) to expand $(s + 4)(s + 6)$ and see if it equals $s^2 + 10s + 24$:

• First: $s * s = s^2$
• Outer: $s * 6 = 6s$
• Inner: $4 * s = 4s$
• Last: $4 * 6 = 24$

Adding these together, we get $s^2 + 6s + 4s + 24 = s^2 + 10s + 24$. So, our factoring is correct! Factoring a quadratic expression is like solving a puzzle – you need to find the right pieces that fit together to form the original expression. Remember, the key is to find two numbers that satisfy both the multiplication and addition conditions. Once you get the hang of it, you'll be factoring quadratics like a pro!

4. The Complete Factored Form

We're almost there! Now that we've factored the quadratic expression $s^2 + 10s + 24$ into $(s + 4)(s + 6)$, we need to remember the GCF we factored out earlier. The complete factored form of the original expression $3s^4 + 30s^3 + 72s^2$ is $3s^2(s + 4)(s + 6)$.

This is the final answer! We've successfully broken down the original expression into its simplest factors. To recap, we first identified and factored out the GCF, which was $3s^2$. Then, we factored the remaining quadratic expression $s^2 + 10s + 24$ into $(s + 4)(s + 6)$. Combining these steps, we arrived at the complete factored form: $3s^2(s + 4)(s + 6)$.

Always remember to include the GCF in your final answer. It's a common mistake to forget about it, but it's an essential part of the factored expression. The complete factored form allows us to easily identify the roots of the equation if we were to set the expression equal to zero. It also simplifies the expression for further algebraic manipulations. So, next time you're faced with a factoring problem, remember to follow these steps: identify the GCF, factor it out, factor the remaining expression, and combine everything for the complete factored form. You've got this!

Factoring is a fundamental skill in algebra, and mastering it will open doors to more advanced topics. Keep practicing, and you'll become a factoring master in no time. Good luck, and happy factoring!