Equivalent Expressions Factoring 10x^2y + 25x^2

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In mathematics, equivalent expressions are expressions that may look different but are equal in value for all possible values of the variables involved. Identifying equivalent expressions is a fundamental skill in algebra, crucial for simplifying equations, solving problems, and understanding mathematical relationships. This article delves into the process of finding an expression equivalent to the given expression, 10x^2y + 25x^2, through factorization. We'll break down the steps, explore common factoring techniques, and clarify why one of the provided options correctly represents the equivalent form.

Factoring: The Key to Equivalent Expressions

Factoring is the process of breaking down an expression into a product of its factors. In the context of algebraic expressions, this often involves identifying common factors among the terms and extracting them. This technique is essential for simplifying expressions, solving equations, and, as in our case, identifying equivalent expressions.

When we look at the expression 10x^2y + 25x^2, the goal is to find the greatest common factor (GCF) of the terms. The GCF is the largest factor that divides evenly into all terms in the expression. By factoring out the GCF, we can rewrite the expression in a simpler, equivalent form. For this expression, both terms contain a multiple of 5 and the variable x raised to the power of 2. Therefore, we look for these common components to simplify the expression effectively.

To factor the expression 10x^2y + 25x^2, we first identify the common factors in the coefficients (10 and 25) and the variables (x^2y and x^2). The greatest common factor of 10 and 25 is 5. Both terms also contain x^2. Thus, the greatest common factor of the entire expression is 5x^2. Factoring out 5x^2 involves dividing each term in the expression by 5x^2 and writing the expression as a product of 5x^2 and the resulting quotient. This process not only simplifies the expression but also reveals its underlying structure, making it easier to manipulate and understand in various mathematical contexts.

Step-by-Step Factoring of 10x^2y + 25x^2

  1. Identify the Greatest Common Factor (GCF):

    • The coefficients are 10 and 25. The GCF of 10 and 25 is 5.
    • Both terms contain x^2. The first term has 'y' while the second does not, so 'y' is not part of the GCF.
    • Therefore, the GCF of the expression is 5x^2.

  2. Factor out the GCF:

    • Divide each term in the expression by the GCF 5x^2:
      • (10x^2y) / (5x^2) = 2y
      • (25x^2) / (5x^2) = 5

  3. Rewrite the expression:

    • Place the GCF outside the parentheses and the results of the division inside the parentheses:
      • 5x^2(2y + 5)

Through this systematic approach, we transform the original expression into a factored form that highlights the common elements and simplifies the structure. Each step in the factoring process is crucial to ensure accuracy and clarity, ultimately leading to the correct equivalent expression. Factoring not only simplifies the expression but also reveals its components, which is essential for solving equations and understanding algebraic relationships.

Analyzing the Options

Now that we've factored the expression 10x^2y + 25x^2 and arrived at the equivalent form 5x^2(2y + 5), we can evaluate the given options to determine which one matches our result.

Let's revisit the provided options:

  • A. 5x^2(2y + 5)
  • B. 5x^2y(5 + 20y)
  • C. 10xy(x + 15y)
  • D. 10x^2(y + 25)

By comparing each option with our factored expression, we can see that option A, 5x^2(2y + 5), exactly matches the result of our factoring process. Options B, C, and D, on the other hand, do not match the correct factored form. These options either have incorrect coefficients, incorrect variable terms, or both, indicating that they are not equivalent to the original expression.

To further illustrate this, we can distribute the terms in each option to see if they match the original expression 10x^2y + 25x^2.

  • For option A, distributing 5x^2 into (2y + 5) yields 10x^2y + 25x^2, which is the original expression.
  • For option B, distributing 5x^2y into (5 + 20y) would result in 25x^2y + 100x2y2, which is not equivalent to the original expression.
  • For option C, distributing 10xy into (x + 15y) results in 10x^2y + 150xy^2, which is also not equivalent to the original expression.
  • Lastly, for option D, distributing 10x^2 into (y + 25) gives 10x^2y + 250x^2, which differs from the original expression.

This comparative analysis confirms that only option A correctly represents the factored form of the original expression, making it the sole equivalent expression among the given choices.

Detailed Explanation of the Correct Option (A)

Option A, 5x^2(2y + 5), is the correct equivalent expression for 10x^2y + 25x^2 because it accurately represents the result of factoring out the greatest common factor (GCF). As we determined earlier, the GCF of the expression is 5x^2. Factoring out 5x^2 from each term involves dividing each term by 5x^2 and placing the result inside the parentheses, with 5x^2 outside as a common factor.

To reiterate the process:

  • Dividing 10x^2y by 5x^2 gives 2y.
  • Dividing 25x^2 by 5x^2 gives 5.

Combining these results, we get the factored form 5x^2(2y + 5).

To verify that this is indeed equivalent to the original expression, we can distribute the 5x^2 back into the parentheses:

  • 5x^2 * 2y = 10x^2y
  • 5x^2 * 5 = 25x^2

Adding these terms together, we have 10x^2y + 25x^2, which matches the original expression. This confirms that 5x^2(2y + 5) is the correct equivalent form.

The correctness of option A highlights the importance of accurate GCF identification and factoring techniques. It demonstrates how breaking down an expression into its fundamental components can simplify and clarify its structure, making it easier to manipulate and understand in various algebraic contexts. Understanding these principles is crucial for success in algebra and beyond.

Why the Other Options Are Incorrect

To further clarify why option A is the only correct answer, let's delve into why options B, C, and D are incorrect. Each incorrect option demonstrates a misunderstanding or misapplication of factoring principles, resulting in expressions that are not equivalent to the original.

Option B: 5x^2y(5 + 20y)

This option incorrectly factors out 5x^2y from the expression. While 5x^2 is a common factor, including 'y' in the factored term is a mistake because the term 25x^2 in the original expression does not contain 'y'. Factoring out 'y' from a term that doesn't have it is a fundamental error in factoring. Additionally, the terms inside the parentheses (5 + 20y) are also incorrect. If we distribute 5x^2y into (5 + 20y), we get 25x^2y + 100x2y2, which is significantly different from the original expression 10x^2y + 25x^2.

Option C: 10xy(x + 15y)

This option makes a different error by factoring out 10xy. While 10 is a factor of the coefficient of the first term (10x^2y), it is not a factor of the coefficient of the second term (25x^2). The greatest common factor of 10 and 25 is 5, not 10. Additionally, the terms inside the parentheses (x + 15y) are incorrect. Distributing 10xy into (x + 15y) yields 10x^2y + 150xy^2, which does not match the original expression. This option demonstrates a misunderstanding of how to identify the greatest common factor and properly factor an expression.

Option D: 10x^2(y + 25)

In this option, 10x^2 is factored out. While x^2 is a common factor, using 10 as the numerical factor is incorrect because, as mentioned earlier, 10 is not a factor of 25. The GCF of 10 and 25 is 5. Additionally, the term inside the parentheses, (y + 25), is incorrect. Distributing 10x^2 into (y + 25) results in 10x^2y + 250x^2, which is not equivalent to the original expression 10x^2y + 25x^2. This error arises from not correctly identifying the greatest common factor and misapplying the distributive property.

Conclusion

In conclusion, identifying equivalent expressions is a critical skill in algebra, essential for simplifying equations and solving mathematical problems. In the expression 10x^2y + 25x^2, we found that option A, 5x^2(2y + 5), is the correct equivalent expression. This was achieved by systematically factoring out the greatest common factor (GCF), which is 5x^2, from both terms in the original expression.

We also analyzed why the other options were incorrect, highlighting common errors in factoring, such as misidentifying the GCF or incorrectly distributing terms. These errors serve as valuable lessons in avoiding pitfalls when simplifying algebraic expressions.

Mastering factoring techniques and understanding the concept of equivalent expressions not only enhances algebraic skills but also builds a strong foundation for more advanced mathematical concepts. By practicing these techniques and paying close attention to detail, one can confidently navigate algebraic manipulations and problem-solving.