Simplify $-5y^2(-5y^2-8y+2)$

Let's dive into simplifying algebraic expressions, specifically focusing on the problem: $-5 y^2 ext{ }(-5 y^2-8 y+2)$. Simplifying expressions like this is a fundamental skill in mathematics that helps us work with more complex equations and understand mathematical relationships more clearly. When we simplify, we're essentially rewriting an expression in its most compact and understandable form without changing its overall value. This is crucial in algebra because it makes it easier to solve equations, analyze functions, and perform further mathematical operations. Think of it like tidying up a messy room; you're not getting rid of anything, just organizing it so it's easier to navigate and use. The expression we're looking at involves multiplying a monomial (a single term, $-5y^2$) by a polynomial (an expression with multiple terms, $-5y^2-8y+2$). The distributive property is our key tool here, allowing us to multiply the term outside the parentheses by each term inside the parentheses individually. This process breaks down a complex multiplication into a series of simpler ones, making the entire task much more manageable and less prone to errors. We'll go through each step methodically, ensuring that we handle the signs and exponents correctly, which are common areas where mistakes can occur. By understanding the underlying principles of exponent rules and distribution, you'll find that simplifying such expressions becomes a straightforward and even satisfying process. This skill is not just for math class; it's a building block for many areas in science, engineering, and even finance where quantitative analysis is essential.

Understanding the Distributive Property

The distributive property is the cornerstone of simplifying expressions like $-5 y^2 ext{ }(-5 y^2-8 y+2)$. This property states that for any numbers a, b, and c, the equation $a(b + c) = ab + ac$ holds true. In simpler terms, when you multiply a number by a sum (or difference), you can distribute the multiplication to each term inside the parentheses. In our problem, the 'a' is $-5y^2$, and the terms inside the parentheses are 'b' ($-5y^2$), 'c' ($-8y$), and 'd' ($+2$). So, we will multiply $-5y^2$ by each of these terms one by one. It's important to remember the rules of multiplying exponents: when you multiply terms with the same base, you add their exponents. For example, $y^2 imes y^2 = y^{2+2} = y^4$. Also, pay close attention to the signs. Multiplying two negative numbers results in a positive number, while multiplying a negative number by a positive number results in a negative number. These are the fundamental rules that will guide us through each step of the simplification process. Mastering the distributive property and exponent rules will not only help you solve this particular problem but also build a strong foundation for tackling more intricate algebraic manipulations. This method ensures that we systematically break down the problem, reducing the chance of errors and leading to a clear, simplified final answer. We're essentially unboxing the multiplication, applying the multiplier to every item within the box.

Step-by-Step Simplification

Let's break down the simplification of $-5 y^2 ext{ }(-5 y^2-8 y+2)$ step by step. Our goal is to apply the distributive property: multiply $-5y^2$ by each term inside the parentheses.

Step 1: Multiply $-5y^2$ by $-5y^2$.

• $-5y^2 imes -5y^2$
  • First, multiply the coefficients: $-5 imes -5 = 25$ (a negative times a negative is positive).
  • Next, multiply the variables: $y^2 imes y^2$. Using the exponent rule for multiplication (add the exponents), we get $y^{2+2} = y^4$.
  • Combining these, we get $25y^4$. This is the first term of our simplified expression.

Step 2: Multiply $-5y^2$ by $-8y$.

• $-5y^2 imes -8y$
  • Multiply the coefficients: $-5 imes -8 = 40$ (again, negative times negative is positive).
  • Multiply the variables: $y^2 imes y$. Remember that $y$ is the same as $y^1$. So, $y^2 imes y^1 = y^{2+1} = y^3$.
  • Combining these, we get $40y^3$. This is the second term of our simplified expression.

Step 3: Multiply $-5y^2$ by $+2$.

• $-5y^2 imes +2$
  • Multiply the coefficients: $-5 imes 2 = -10$ (a negative times a positive is negative).
  • Multiply the variables: $y^2$. Since there's no variable in the second term, $y^2$ remains as it is.
  • Combining these, we get $-10y^2$. This is the third term of our simplified expression.

Step 4: Combine all the terms.

Now, we put all the results from the previous steps together:

$25y^4 + 40y^3 - 10y^2$

These terms cannot be combined further because they have different variable parts (different powers of y). Therefore, the fully simplified expression is $25y^4 + 40y^3 - 10y^2$. Each step was performed carefully, applying the rules of multiplication for signs and exponents. This systematic approach ensures accuracy in algebraic simplification.

Final Answer and Verification

After meticulously following the steps using the distributive property, we have arrived at the simplified form of the expression $-5 y^2 ext{ }(-5 y^2-8 y+2)$. The final simplified expression is $25y^4 + 40y^3 - 10y^2$. We've ensured that each term inside the parentheses was multiplied by the monomial outside, paying close attention to the rules of multiplying coefficients and adding exponents when bases are the same. The signs were also carefully managed: a negative multiplied by a negative yields a positive, and a negative multiplied by a positive yields a negative. The resulting terms, $25y^4$, $40y^3$, and $-10y^2$, are unlike terms because they have different powers of the variable $y$. Therefore, they cannot be combined further. This means our expression is in its simplest form.

To briefly verify our work, we can think about plugging in a simple value for $y$, say $y=1$.

• Original expression: $-5(1)^2(-5(1)^2 - 8(1) + 2) = -5(1)(-5 - 8 + 2) = -5(-11) = 55$.
• Simplified expression: $25(1)^4 + 40(1)^3 - 10(1)^2 = 25(1) + 40(1) - 10(1) = 25 + 40 - 10 = 65 - 10 = 55$.

Since both expressions evaluate to the same value, it gives us confidence that our simplification is correct. This kind of verification, while not a formal proof, is a very useful technique for catching errors in algebraic manipulations. Understanding how to simplify expressions like this is a key skill that unlocks the door to more advanced mathematical concepts. It's about making complex things manageable and revealing the underlying structure.

For further exploration into algebraic simplification and the distributive property, you can visit resources like Khan Academy, which offers excellent tutorials and practice exercises on these fundamental mathematics topics. Additionally, Math is Fun provides clear explanations and examples that can help solidify your understanding.