Factoring Completely -5x³ + 10x² A Step-by-Step Guide

Factoring polynomials is a fundamental skill in algebra, and mastering it opens doors to solving various mathematical problems. In this comprehensive guide, we will delve into the process of factoring the polynomial $-5x^3 + 10x^2$ completely. We will explore the steps involved, understand the underlying concepts, and arrive at the correct factored form. This detailed explanation aims to provide a clear and concise understanding of factoring, empowering you to tackle similar problems with confidence.

Understanding Factoring

Before we dive into the specifics of our polynomial, let's establish a solid understanding of what factoring entails. Factoring is essentially the reverse process of expanding or multiplying expressions. When we factor, we aim to break down a polynomial into its constituent factors, which are expressions that, when multiplied together, yield the original polynomial. These factors can be numbers, variables, or even other polynomials. Factoring helps us simplify expressions, solve equations, and gain insights into the behavior of mathematical functions.

Think of it like this: if you have a number like 12, you can factor it into 3 x 4, or 2 x 6, or even 2 x 2 x 3. Similarly, with polynomials, we are looking for the building blocks that multiply to give us the original expression. The goal is to find the greatest common factor (GCF) and express the polynomial as a product of this GCF and another polynomial. Factoring completely means breaking down the polynomial into its simplest possible factors, leaving no further room for factorization. This is crucial for solving equations and simplifying expressions in algebra and beyond. The process not only simplifies complex equations but also reveals the structure and properties of the underlying mathematical relationships.

Identifying the Greatest Common Factor (GCF)

The first and most crucial step in factoring any polynomial is identifying the greatest common factor (GCF). The GCF is the largest factor that divides evenly into all terms of the polynomial. It's like finding the biggest piece that fits into all parts of a puzzle. To find the GCF, we look at both the coefficients (the numbers in front of the variables) and the variables themselves.

For the coefficients, we find the largest number that divides all of them without leaving a remainder. For the variables, we identify the variable with the smallest exponent that appears in all terms. Let's consider our polynomial, $-5x^3 + 10x^2$. The coefficients are -5 and 10. The largest number that divides both -5 and 10 is 5. Now, considering the signs, we can also factor out -5 to make the leading coefficient positive inside the parenthesis. This is a common practice as it often simplifies further steps. For the variables, we have $x^3$ and $x^2$. The smallest exponent is 2, so the variable part of the GCF is $x^2$. Combining these, our GCF is $-5x^2$. This means that $-5x^2$ is the largest expression that can be divided out of both terms in our polynomial, simplifying it and paving the way for further factorization if necessary. This step is fundamental because it simplifies the original expression, making subsequent factorization steps easier to manage.

Factoring out the GCF

Once we've identified the GCF, the next step is to factor it out of the polynomial. This involves dividing each term of the polynomial by the GCF and writing the result in parentheses. It's like reverse distribution – we're pulling out a common factor instead of multiplying it in. In our case, the GCF is $-5x^2$, and our polynomial is $-5x^3 + 10x^2$. We divide each term by $-5x^2$:

• $-5x^3 / -5x^2 = x$
• $10x^2 / -5x^2 = -2$

Now, we write the GCF outside the parentheses and the results of the division inside the parentheses:

$-5x^2(x - 2)$.

This expression represents the factored form of the original polynomial. We have successfully extracted the greatest common factor, which simplifies the polynomial and makes it easier to analyze. This step is crucial in solving equations and understanding the behavior of polynomial functions. By factoring out the GCF, we reduce the degree of the polynomial inside the parentheses, often leading to a more manageable expression for further factorization or solving.

Checking Your Work

An essential step in factoring is to check your work. The easiest way to do this is to distribute the GCF back into the parentheses. If you arrive back at the original polynomial, you've factored correctly. It's like retracing your steps to make sure you haven't made any errors. Let's check our factored form, $-5x^2(x - 2)$. We distribute $-5x^2$ to both terms inside the parentheses:

• $-5x^2 * x = -5x^3$
• $-5x^2 * -2 = 10x^2$

Combining these, we get $-5x^3 + 10x^2$, which is our original polynomial. This confirms that our factoring is correct. Checking your work is a critical habit to develop in mathematics, as it ensures accuracy and prevents errors from propagating through subsequent steps. It provides confidence in your solution and helps reinforce your understanding of the factoring process. This step is particularly important in more complex problems where the chances of making a mistake are higher.

Analyzing the Options

Now that we have factored the polynomial completely, let's examine the given options and identify the correct answer. Understanding why certain options are incorrect is just as crucial as identifying the correct one. This process deepens your understanding of factoring and enhances your problem-solving skills.

The original question provided the following options:

A. Prime

B. $-x(5x^2 - 10x)$

C. $5x(-x^2 + 2x)$

D. $-5x^2(x - 2)$

Let’s break down each option:

• Option A: Prime A polynomial is considered prime if it cannot be factored further. Our factored form, $-5x^2(x - 2)$, clearly shows that the polynomial can be factored, so this option is incorrect. Recognizing prime polynomials is an important skill, but in this case, the polynomial is factorable.

• Option B: $-x(5x^2 - 10x)$ This option does have a common factor of $-x$ but doesn't completely factor out the greatest common factor. Factoring out $-x$ leaves $5x^2 - 10x$ inside the parentheses, which can be further factored. Therefore, this option is partially factored but not completely, making it incorrect. The ability to recognize when a polynomial is not fully factored is crucial for solving more complex problems.

• Option C: $5x(-x^2 + 2x)$ Similar to option B, this option has a common factor of $5x$ factored out, but the expression inside the parentheses, $-x^2 + 2x$, still has a common factor of $x$. This indicates that the polynomial is not fully factored. Therefore, this option is incorrect. Recognizing the potential for further factorization is a key aspect of mastering factoring techniques.

• Option D: $-5x^2(x - 2)$ This is the factored form we obtained by factoring out the GCF, $-5x^2$, from the original polynomial. The expression inside the parentheses, $x - 2$, cannot be factored further. This option represents the complete factorization of the polynomial. Therefore, this is the correct answer.

By systematically analyzing each option, we reinforce our understanding of complete factorization and avoid common pitfalls. This step is invaluable in developing a strong foundation in algebra and beyond.

The Correct Answer

After carefully factoring the polynomial $-5x^3 + 10x^2$ and analyzing the given options, we can confidently conclude that the correct answer is:

D. $-5x^2(x - 2)$

This expression represents the complete factorization of the polynomial. We have successfully identified the greatest common factor, factored it out, and verified our result. This process highlights the importance of understanding the principles of factoring and applying them systematically to arrive at the correct solution. The ability to factor polynomials completely is a fundamental skill that underpins many areas of mathematics, from solving equations to simplifying complex expressions. Mastering this skill empowers you to tackle a wide range of problems with confidence and precision. Remember, practice is key to honing your factoring skills, so continue to work through various examples to solidify your understanding.

Conclusion

Factoring polynomials is a crucial skill in algebra, with applications extending to various mathematical disciplines. In this guide, we meticulously factored the polynomial $-5x^3 + 10x^2$, demonstrating the step-by-step process of identifying the greatest common factor, factoring it out, and verifying the result. We also analyzed different options to understand why some factorizations are incomplete and why the correct answer, $-5x^2(x - 2)$, represents the complete factorization. This comprehensive approach not only solves the specific problem but also equips you with a deeper understanding of factoring principles. By mastering these techniques, you will be well-prepared to tackle more complex factoring problems and excel in your mathematical journey. Remember, consistent practice and a systematic approach are essential for success in algebra. Embrace the challenge, and you'll find that factoring becomes a powerful tool in your mathematical toolkit.