Finding The Product Of -3 And A Trinomial Factoring And Algebraic Manipulation

In this article, we will delve into the process of finding the product of -3 and a trinomial, a fundamental concept in algebra. Specifically, we'll address the question: What is the correct product when you factor out (-3) from a trinomial, given that you obtained (-3)(x^2) + (-3)(-7x) + (-3)(10)? We'll explore the underlying principles, step-by-step calculations, and the reasoning behind the correct answer. This comprehensive guide aims to provide clarity and a deep understanding of this algebraic manipulation. Let's begin by dissecting the problem and understanding the core concepts involved. We'll then systematically work through the solution, ensuring that every step is clear and well-explained. This approach will not only help in solving this particular problem but also in tackling similar algebraic challenges in the future. Our goal is to make algebra accessible and understandable, fostering a strong foundation for more advanced mathematical concepts. Remember, mathematics is not just about finding the right answer; it's about understanding the process and the logic behind it. So, let's embark on this journey of algebraic exploration and discovery.

Deconstructing the Problem: Factoring Out -3

The initial expression we are given is (-3)(x^2) + (-3)(-7x) + (-3)(10). This represents a trinomial where each term has been multiplied by -3. The task at hand is to reverse this process, effectively factoring out the -3. Factoring is a crucial skill in algebra, allowing us to simplify expressions and solve equations more efficiently. It involves identifying common factors in each term and extracting them to rewrite the expression in a more manageable form. In this case, the common factor is clearly -3. When we factor out -3, we are essentially dividing each term by -3 and placing the -3 outside the parentheses. This process relies on the distributive property of multiplication over addition and subtraction, which states that a(b + c) = ab + ac. By understanding this property, we can confidently manipulate algebraic expressions and arrive at the correct solution. The key is to pay close attention to the signs and ensure that each term is handled correctly. Factoring out negative numbers can sometimes be tricky, but with a clear understanding of the rules and careful application, it becomes a straightforward process. So, let's proceed to factor out the -3 and see what trinomial remains within the parentheses. This will bring us closer to identifying the correct product.

Step-by-Step Factoring: Unveiling the Trinomial

To factor out -3 from the expression (-3)(x^2) + (-3)(-7x) + (-3)(10), we divide each term by -3 and place the -3 outside the parentheses. This can be visualized as follows:

-3( (x^2) + (-7x) + (10) )

Now, let's simplify the terms inside the parentheses:

• (x^2) divided by -3 becomes x^2 (since we're factoring out the -3, we're left with the original term inside). However, it's more accurate to say we are factoring out -3 from -3x^2, which leaves us with x^2.
• (-7x) divided by -3 becomes -7x (similarly, factoring -3 from -3 * -7x leaves us with -7x inside the parenthesis).
• (10) divided by -3 becomes 10 (factoring -3 from -3 * 10 leaves 10 inside the parenthesis).

Therefore, the expression inside the parentheses simplifies to (x^2 - 7x + 10). This trinomial is the result of factoring out -3 from the original expression. It's crucial to remember the signs when performing these operations. A negative divided by a negative results in a positive, and a negative divided by a positive results in a negative. By meticulously following these rules, we can avoid common errors and ensure the accuracy of our calculations. Factoring is a fundamental skill in algebra, and mastering it is essential for solving more complex problems. So, let's move on to identifying the correct option from the given choices, now that we have successfully factored out the -3 and simplified the trinomial.

Identifying the Correct Product: Matching the Trinomial

Now that we have factored out -3 and obtained the expression -3(x^2 - 7x + 10), we need to match this result with the given options. The options provided are:

• A. (-3)(-x^2 + 7x + (-10))
• B. (-3)(x^2 - 7x + 10)
• C. (-3)(x^2 + 7x + 10)

By comparing our result, -3(x^2 - 7x + 10), with the options, we can clearly see that option B, (-3)(x^2 - 7x + 10), matches our factored expression exactly. This confirms that option B is the correct answer. The other options differ in the signs of the terms within the parentheses. Option A has the signs reversed for all terms, and option C has the sign of the middle term incorrect. This highlights the importance of careful attention to signs when factoring and simplifying algebraic expressions. A small error in the sign can lead to a completely different result. Therefore, it's always a good practice to double-check your work and ensure that each step is performed accurately. In this case, by meticulously factoring out -3 and comparing the result with the options, we have confidently identified the correct product. Let's now discuss why the other options are incorrect to further solidify our understanding.

Why Other Options Are Incorrect: Understanding Sign Errors

To further solidify our understanding, let's analyze why options A and C are incorrect. This will help us understand the common pitfalls in factoring and manipulating algebraic expressions. Option A, (-3)(-x^2 + 7x + (-10)), is incorrect because the signs of all the terms inside the parentheses are reversed compared to the correct trinomial. This would only be the case if we had factored out 3 instead of -3 from the expression -3x^2 + 21x - 30. Factoring out a positive 3 would result in 3(-x^2 + 7x - 10), which is similar to option A but with a positive 3 outside the parentheses. The crucial difference lies in the sign of the factor. Option C, (-3)(x^2 + 7x + 10), is incorrect because the sign of the middle term, the 7x term, is positive instead of negative. This error could arise from incorrectly dividing 21x by -3, resulting in +7x instead of -7x. It highlights the importance of carefully tracking the signs during division and multiplication. These incorrect options underscore the significance of paying close attention to the signs when factoring. A simple sign error can completely change the expression and lead to a wrong answer. By understanding why these options are incorrect, we reinforce our understanding of the correct factoring process and the importance of meticulous calculations. Now, let's summarize the entire process and highlight the key takeaways.

Conclusion: Mastering the Art of Factoring

In conclusion, we have successfully determined the correct product of -3 and the remaining trinomial after factoring. The correct answer is (-3)(x^2 - 7x + 10), which corresponds to option B. We arrived at this solution by meticulously factoring out -3 from the expression (-3)(x^2) + (-3)(-7x) + (-3)(10), step by step. This process involved dividing each term by -3 and placing the -3 outside the parentheses. We also discussed why the other options were incorrect, emphasizing the importance of paying close attention to signs during factoring and algebraic manipulations. Sign errors are a common pitfall in algebra, and avoiding them requires careful attention to detail. Factoring is a fundamental skill in algebra, and mastering it is crucial for solving more complex problems. It allows us to simplify expressions, solve equations, and gain a deeper understanding of mathematical relationships. By practicing factoring and understanding the underlying principles, we can build a strong foundation for advanced mathematical concepts. Remember, mathematics is not just about memorizing formulas; it's about understanding the process and the logic behind it. We hope this comprehensive guide has provided clarity and a deeper understanding of factoring and algebraic manipulations. Keep practicing, and you'll become more confident and proficient in solving algebraic problems.