Factoring Polynomials: A Step-by-Step Guide

Hey guys! Factoring polynomials can seem like a daunting task, but trust me, it's a crucial skill in mathematics. Let's break down how to factor the polynomial $5 c^5+60 c^4+180 c^3$ completely. We'll go through each step to make sure you understand the process. By the end of this guide, you'll be able to tackle similar problems with confidence. Factoring polynomials is essential for simplifying expressions, solving equations, and understanding more advanced mathematical concepts. So, let’s dive in and make factoring a breeze!

1. Identify the Greatest Common Factor (GCF)

The first step in factoring any polynomial is to identify the greatest common factor (GCF) of all the terms. In our polynomial, $5 c^5+60 c^4+180 c^3$, we need to find the largest factor that divides evenly into all three terms. Look at the coefficients (5, 60, and 180) and the variable terms ($c^5$, $c^4$, and $c^3$). For the coefficients, the GCF is 5 since it’s the largest number that divides 5, 60, and 180. Now, let’s consider the variable part. We have $c^5$, $c^4$, and $c^3$. The GCF here is the lowest power of $c$ present in all terms, which is $c^3$. Therefore, the overall GCF for the polynomial is $5c^3$. This means we can factor out $5c^3$ from each term. Factoring out the GCF simplifies the polynomial and makes it easier to work with. It’s like finding the basic building block that all terms have in common. This step is crucial because it reduces the complexity of the expression, allowing us to apply further factoring techniques more easily. Always start by looking for the GCF; it's a game-changer!

2. Factor Out the GCF

Now that we've identified the GCF as $5c^3$, let's factor it out of the polynomial $5 c^5+60 c^4+180 c^3$. To do this, we divide each term in the polynomial by the GCF. So, we divide $5c^5$ by $5c^3$, $60c^4$ by $5c^3$, and $180c^3$ by $5c^3$. When we divide $5c^5$ by $5c^3$, we get $c^2$ (because $c^5 / c^3 = c^{5-3} = c^2$). Next, dividing $60c^4$ by $5c^3$ gives us $12c$ (since $60/5 = 12$ and $c^4 / c^3 = c^{4-3} = c$). Finally, $180c^3$ divided by $5c^3$ equals 36 (as $180/5 = 36$ and $c^3 / c^3 = 1$). Putting it all together, we have $5c^3(c^2 + 12c + 36)$. By factoring out the GCF, we've transformed the original polynomial into a product of the GCF and a simpler quadratic expression. This makes the next steps of factoring much more manageable. Remember, factoring out the GCF is like peeling away the outer layers to reveal the core structure of the polynomial. It's a fundamental step that sets the stage for further simplification and factoring techniques.

3. Recognize the Perfect Square Trinomial

After factoring out the GCF, we're left with the expression $5 c^3(c^2+12 c+36)$. Now, let's focus on the quadratic expression inside the parentheses: $c^2 + 12c + 36$. We need to determine if this trinomial can be factored further. One common pattern to look for is a perfect square trinomial. A perfect square trinomial is a trinomial that can be written in the form $(a + b)^2$ or $(a - b)^2$. To identify a perfect square trinomial, check if the first and last terms are perfect squares and if the middle term is twice the product of the square roots of the first and last terms. In our case, the first term is $c^2$, which is a perfect square (its square root is $c$). The last term is 36, which is also a perfect square (its square root is 6). Now, let's check the middle term. The middle term is $12c$. If we take twice the product of the square roots of the first and last terms, we get $2 * c * 6 = 12c$, which matches our middle term. This confirms that $c^2 + 12c + 36$ is indeed a perfect square trinomial. Recognizing this pattern is super helpful because it allows us to quickly factor the trinomial into a squared binomial. This simplifies the factoring process and gets us closer to the completely factored form of the original polynomial. So, always be on the lookout for perfect square trinomials; they're a common pattern that can make factoring much easier!

4. Factor the Perfect Square Trinomial

Now that we've identified $c^2 + 12c + 36$ as a perfect square trinomial, let's factor it. A perfect square trinomial of the form $a^2 + 2ab + b^2$ can be factored as $(a + b)^2$. In our trinomial, $c^2 + 12c + 36$, we can see that $a$ corresponds to $c$ and $b$ corresponds to 6 (since $6^2 = 36$). The middle term, $12c$, fits the pattern $2ab$ because $2 * c * 6 = 12c$. Therefore, we can factor $c^2 + 12c + 36$ as $(c + 6)^2$. This means we're expressing the trinomial as a binomial squared, which is a more simplified form. Substituting this back into our expression from step 2, we have $5c^3(c + 6)^2$. Factoring a perfect square trinomial is like fitting puzzle pieces together; once you recognize the pattern, the factorization becomes straightforward. By factoring the trinomial, we've taken another step towards completely factoring the original polynomial. Remember, the goal is to break down the polynomial into its simplest factors, and recognizing patterns like perfect square trinomials makes the process much more efficient and less prone to errors.

5. Write the Completely Factored Form

We've done the hard work, guys! Now, let's put it all together. We started with the polynomial $5 c^5+60 c^4+180 c^3$. In step 1, we identified the greatest common factor (GCF) as $5c^3$. In step 2, we factored out the GCF, giving us $5c^3(c^2 + 12c + 36)$. Then, in steps 3 and 4, we recognized that the quadratic expression $c^2 + 12c + 36$ is a perfect square trinomial and factored it as $(c + 6)^2$. Now, we combine all these steps to write the completely factored form of the polynomial. We have the GCF, $5c^3$, and the factored trinomial, $(c + 6)^2$. So, the completely factored form of $5 c^5+60 c^4+180 c^3$ is $5 c^3(c+6)^2$. This is our final answer! We've broken down the polynomial into its simplest factors, and we can't factor it any further. Writing the completely factored form is like putting the final piece in a jigsaw puzzle; it's the satisfying conclusion to our factoring journey. Make sure to double-check your work and ensure that you've factored out all common factors and simplified the expression as much as possible. Great job, everyone! You've successfully factored the polynomial completely.

6. Final Answer

So, after all our hard work, the completely factored form of the polynomial $5 c^5+60 c^4+180 c^3$ is: C. $5 c^3(c+6)^2$. Awesome! We took it step-by-step, guys. Factoring out the GCF first made the rest easier, and spotting that perfect square trinomial was key. Remember, practice makes perfect, so keep at it! Factoring can seem tough at first, but with time, you'll nail it. If you get stuck, just go back to the basics: look for the GCF, check for patterns, and break it down. You got this! This answer matches option C, which is the correct factorization. Factoring polynomials can be challenging, but by following these steps, you can simplify complex expressions and solve various mathematical problems. This skill is crucial in algebra and calculus, making it a valuable tool in your mathematical journey. Keep practicing, and you'll become a factoring pro in no time!