Factoring The Expression 3xy² + 15x⁴y³ - 10x²y A Step By Step Guide
Factoring expressions is a fundamental skill in algebra, enabling us to simplify complex equations and solve for unknowns. In this article, we will explore the process of factoring the algebraic expression 3xy² + 15x⁴y³ - 10x²y. We will break down each step, identify the common factors, and arrive at the factored form of the expression. Understanding factoring not only helps in simplifying expressions but also lays the groundwork for solving higher-level algebraic problems. The ability to factor correctly is crucial for success in various areas of mathematics, making it an indispensable skill for students and professionals alike.
Identifying the Greatest Common Factor (GCF)
In tackling the problem of factoring 3xy² + 15x⁴y³ - 10x²y, the initial and most crucial step involves identifying the Greatest Common Factor (GCF) of the terms. The GCF is the largest factor that divides evenly into all terms of the expression. This foundational step simplifies the expression by extracting the common elements, making subsequent steps more manageable. To find the GCF, we first examine the coefficients and then the variables present in each term. In our given expression, the coefficients are 3, 15, and -10. The GCF of these numbers is the largest number that can divide each of them without leaving a remainder. By inspection, we see that the GCF of 3, 15, and -10 is 1, as they do not share any common factors other than 1.
Next, we turn our attention to the variables. The expression contains the variables x and y, each raised to different powers in the terms. To find the GCF of the variables, we consider the lowest power of each variable that appears in all terms. For the variable x, we have x, x⁴, and x². The lowest power of x is x¹, or simply x. For the variable y, we have y², y³, and y. The lowest power of y is y¹, or simply y. Therefore, the GCF of the variables is xy. Combining the GCF of the coefficients (which is 1) with the GCF of the variables (xy), we determine that the overall GCF of the expression 3xy² + 15x⁴y³ - 10x²y is xy. This identification is pivotal as it allows us to factor out xy from the original expression, setting the stage for simplifying it into a more concise form. This methodical approach ensures that we are accurately extracting the greatest common factor, which is essential for correct factoring.
Factoring Out the GCF
Having successfully identified the Greatest Common Factor (GCF) as xy for the expression 3xy² + 15x⁴y³ - 10x²y, the next critical step is to factor it out. This process involves dividing each term of the original expression by the GCF and writing the result in parentheses. Factoring out the GCF is akin to reversing the distributive property, which makes the expression simpler and easier to work with. We start by dividing the first term, 3xy², by xy. This gives us 3y, as the x terms cancel out (x/x = 1) and one y remains (y²/y = y). Next, we divide the second term, 15x⁴y³, by xy. This results in 15x³y², where x becomes x³ (x⁴/x = x³) and y becomes y² (y³/y = y²). Finally, we divide the third term, -10x²y, by xy, which yields -10x, where x becomes x (x²/x = x) and the y terms cancel out (y/y = 1). After performing these divisions, we gather the quotients inside a parenthesis, ensuring each term retains its original sign. This process leads us to the factored form: xy(3y + 15x³y² - 10x). This factored expression is equivalent to the original expression but is now represented as a product of the GCF and the remaining terms. The act of factoring out the GCF not only simplifies the expression but also reveals its underlying structure, which is beneficial for further algebraic manipulations. By systematically dividing each term by the GCF, we ensure that the factored form is accurate and that no common factors remain within the parentheses. This method provides a clear and concise way to express the original polynomial in a simplified, factored manner.
Verifying the Factored Form
After factoring out the Greatest Common Factor (GCF) from the expression 3xy² + 15x⁴y³ - 10x²y, and arriving at the factored form xy(3y + 15x³y² - 10x), it is crucial to verify the result. Verification ensures that the factored form is indeed equivalent to the original expression. The most straightforward method for verification is to apply the distributive property, which involves multiplying the GCF (xy) by each term inside the parentheses (3y + 15x³y² - 10x). By doing so, we should obtain the original expression if the factoring was performed correctly. Let’s begin by multiplying xy with 3y. This yields 3xy², which matches the first term of the original expression. Next, we multiply xy with 15x³y², resulting in 15x⁴y³. This corresponds to the second term of the original expression. Finally, multiplying xy with -10x gives us -10x²y, which is the third term of the original expression. By combining these results, we have 3xy² + 15x⁴y³ - 10x²y, which is exactly the original expression we started with. This confirms that our factoring process was accurate and that the factored form xy(3y + 15x³y² - 10x) is equivalent to the initial expression. This verification step is an essential practice in algebra, as it helps to catch any errors made during the factoring process. It reinforces the understanding of how factoring and the distributive property are inverse operations, and it provides confidence in the correctness of the solution. This careful checking ensures that the simplified expression is not only simpler but also mathematically sound.
Solution and Answer Options
Having successfully factored the expression 3xy² + 15x⁴y³ - 10x²y, we arrived at the factored form xy(3y + 15x³y² - 10x). Now, let's compare this result with the given answer options to identify the correct one. The options presented are:
A. 3xy(y + 5x³y² - 10x) B. 5x²y²(3x + 3x²y - 2) C. xy(3y + 15x³y² - 10x) D. xy(3xy + 15x⁴y³ - 10xy)
By carefully examining each option, we can see that option C, xy(3y + 15x³y² - 10x), perfectly matches the factored form we derived. This indicates that option C is the correct answer. It is crucial to note that the other options do not match our factored form. Option A, 3xy(y + 5x³y² - 10x), has an extra factor of 3 in front of the xy, which is not present in our factored form. Option B, 5x²y²(3x + 3x²y - 2), has a completely different set of factors and terms inside the parentheses, making it incorrect. Option D, xy(3xy + 15x⁴y³ - 10xy), includes terms inside the parentheses that still contain the common factor xy, indicating that the factoring process was not fully completed. Therefore, the only option that accurately represents the factored form of the original expression is option C. This process of comparing the derived solution with the provided options is a valuable step in problem-solving. It ensures that the correct answer is selected and reinforces the understanding of the factoring process.
Conclusion
In conclusion, we have successfully factored the algebraic expression 3xy² + 15x⁴y³ - 10x²y. The process began with identifying the Greatest Common Factor (GCF), which was determined to be xy. We then factored out the GCF from each term of the expression, resulting in the factored form xy(3y + 15x³y² - 10x). To ensure the accuracy of our factoring, we verified the result by applying the distributive property, multiplying the GCF by the terms inside the parentheses. This verification confirmed that the factored form is indeed equivalent to the original expression. Finally, by comparing our factored form with the given answer options, we identified option C, xy(3y + 15x³y² - 10x), as the correct answer. This exercise underscores the importance of factoring in algebra, which simplifies expressions and aids in solving equations. The methodical approach of identifying the GCF, factoring it out, and verifying the result is crucial for accurate and efficient problem-solving. Factoring is a fundamental skill that not only simplifies algebraic manipulations but also enhances the understanding of mathematical structures. By mastering factoring techniques, students can confidently tackle more complex problems and gain a deeper appreciation for the elegance and precision of mathematics. The ability to factor expressions accurately is a valuable asset in various mathematical contexts and beyond.
