Expanding And Simplifying The Product (-2d^2 + S)(5d^2 - 6s)

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In the realm of mathematics, particularly in algebra, expressions often come in various forms, and understanding how to manipulate and simplify them is crucial. This article delves into the product of two such expressions: (-2d^2 + s)(5d^2 - 6s). We'll break down the process of expanding this product, discuss the underlying principles, and explore the significance of this type of algebraic manipulation. Whether you're a student grappling with algebraic concepts or someone looking to refresh your mathematical skills, this comprehensive guide will provide you with a clear understanding of how to handle such expressions.

Expanding the Product: The Distributive Property

At its core, expanding the product of two expressions like (-2d^2 + s)(5d^2 - 6s) relies on the distributive property. This fundamental principle states that for any numbers or expressions a, b, and c, a(b + c) = ab + ac. In simpler terms, it means that you can multiply a single term by each term within a parenthesis and then add the results. When dealing with two expressions, each containing multiple terms, we extend this principle by distributing each term in the first expression across every term in the second expression. Let's apply this to our example:

(-2d^2 + s)(5d^2 - 6s)

We start by taking the first term of the first expression, -2d^2, and multiplying it by each term in the second expression:

-2d^2 * (5d^2 - 6s) = (-2d^2 * 5d^2) + (-2d^2 * -6s) = -10d^4 + 12d^2s

Next, we take the second term of the first expression, s, and multiply it by each term in the second expression:

s * (5d^2 - 6s) = (s * 5d^2) + (s * -6s) = 5d^2s - 6s^2

Now, we combine the results from these two steps:

(-10d^4 + 12d^2s) + (5d^2s - 6s^2)

This gives us the expanded form of the product, but we're not quite done yet. The next step is to simplify the expression by combining like terms.

Simplifying the Expression: Combining Like Terms

After expanding the product, we often end up with multiple terms that share the same variables and exponents. These are called like terms, and they can be combined to simplify the expression. In our expanded form, -10d^4 + 12d^2s + 5d^2s - 6s^2, we can identify two like terms: 12d^2s and 5d^2s. These terms both have the variables d and s, with d raised to the power of 2 and s raised to the power of 1. To combine them, we simply add their coefficients:

12d^2s + 5d^2s = (12 + 5)d^2s = 17d^2s

Now, we substitute this combined term back into our expression:

-10d^4 + 17d^2s - 6s^2

This is the simplified form of the product (-2d^2 + s)(5d^2 - 6s). We have successfully expanded the product using the distributive property and then simplified the result by combining like terms.

Understanding the Significance of Algebraic Manipulation

The process of expanding and simplifying algebraic expressions is not just a mathematical exercise; it's a fundamental skill with wide-ranging applications. This type of manipulation is crucial in various areas, including:

  • Solving Equations: Many equations require simplification before they can be solved. Expanding products and combining like terms are often necessary steps in isolating the variable and finding its value.
  • Graphing Functions: Understanding the expanded form of an equation can help in visualizing its graph. The coefficients and exponents reveal important information about the function's behavior, such as its intercepts, symmetry, and end behavior.
  • Calculus: In calculus, algebraic manipulation is essential for finding derivatives and integrals. Simplifying expressions can make these operations significantly easier.
  • Physics and Engineering: Many physical laws and engineering formulas involve algebraic expressions. Being able to manipulate these expressions is crucial for solving problems and making predictions.

Moreover, the ability to work with algebraic expressions fosters logical thinking and problem-solving skills that are valuable in many aspects of life. It encourages a systematic approach to complex problems and helps develop analytical abilities.

Common Mistakes and How to Avoid Them

While the process of expanding and simplifying expressions is straightforward, there are some common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure accurate results:

  • Sign Errors: One of the most common mistakes is making errors with signs, especially when dealing with negative terms. Pay close attention to the signs of each term and use parentheses to avoid confusion.
  • Incorrect Distribution: Forgetting to distribute a term to all the terms within the parentheses is another common mistake. Make sure you multiply each term in the first expression by every term in the second expression.
  • Combining Unlike Terms: Only like terms can be combined. Avoid adding or subtracting terms that have different variables or exponents.
  • Order of Operations: Remember to follow the order of operations (PEMDAS/BODMAS). Multiplication should be performed before addition or subtraction.

To avoid these mistakes, it's helpful to be organized and methodical in your approach. Write out each step clearly and double-check your work. Practice is also key to mastering these skills.

Practice Problems

To solidify your understanding, let's work through a couple of practice problems:

Problem 1: Expand and simplify (3x + 2)(x - 4)

Solution:

  1. Distribute the first term: 3x * (x - 4) = 3x^2 - 12x
  2. Distribute the second term: 2 * (x - 4) = 2x - 8
  3. Combine the results: (3x^2 - 12x) + (2x - 8) = 3x^2 - 10x - 8

Problem 2: Expand and simplify (a^2 - b)(2a^2 + 3b)

Solution:

  1. Distribute the first term: a^2 * (2a^2 + 3b) = 2a^4 + 3a^2b
  2. Distribute the second term: -b * (2a^2 + 3b) = -2a^2b - 3b^2
  3. Combine the results: (2a^4 + 3a^2b) + (-2a^2b - 3b^2) = 2a^4 + a^2b - 3b^2

By working through these examples, you can see the process in action and gain confidence in your ability to handle similar problems.

Conclusion

Expanding and simplifying algebraic expressions is a fundamental skill in mathematics with wide-ranging applications. In this article, we've explored the process of expanding the product (-2d^2 + s)(5d^2 - 6s), emphasizing the distributive property and the importance of combining like terms. We've also discussed the significance of algebraic manipulation in various fields and highlighted common mistakes to avoid. By understanding these concepts and practicing regularly, you can develop a strong foundation in algebra and enhance your problem-solving abilities.

Remember, mathematics is a journey of continuous learning and practice. The more you engage with these concepts, the more proficient you'll become. So, keep exploring, keep practicing, and keep expanding your mathematical horizons.