Calculating Polynomial Value: A Step-by-Step Guide

Hey everyone! Today, we're diving into a cool math problem. We're gonna figure out the value of the expression ${y^4 - 3y^3 + 7y^2 - 12}$ when ${y = 2}$. It's like a fun little puzzle, and I'll walk you through it step-by-step. Don't worry, it's easier than it looks! This is a classic example of evaluating a polynomial. Let's get started, shall we?

Understanding the Problem: Plugging in the Value

So, the deal is, we have this expression: ${y^4 - 3y^3 + 7y^2 - 12}$. It's a polynomial, which just means it's made up of terms with variables (in this case, ${y}$) raised to different powers. We also have some constants (like -3, 7, and -12). The problem tells us that ${y = 2}$. This means, wherever we see ${y}$ in our expression, we're gonna replace it with the number 2. Think of it like a substitution – we're swapping out the letter for a number. That's the key idea here. We're not solving for ${y}$; we're evaluating the expression when ${y}$ has a specific value. This concept is fundamental in algebra and appears in a ton of other areas. When we substitute in the number 2 for every appearance of the variable y, we get the following: ${(2)^4 - 3*(2)^3 + 7*(2)^2 - 12}$. See how we've just replaced every y with a 2? Now we just need to work through this expression, remembering the order of operations (PEMDAS/BODMAS) to make sure we get the right answer. This means we must first handle the exponentiation, then multiplication, then addition and subtraction. This approach guarantees a consistent and accurate result.

Now that we understand the process, let's do it! This process is at the heart of algebra, a super useful skill for all sorts of problems. We're going to simplify the expression, following the order of operations, to reveal a single value, which represents the expression's value at y = 2. This helps simplify complex calculations to arrive at a definitive answer. In this case, the value of the polynomial will be a single number. Remember, the fundamental idea is to substitute the given value and then carefully execute the mathematical operations to reach the final answer. Now that you grasp the basics, let's begin the computation, moving step by step to get the correct solution.

Step-by-Step Calculation: Breaking it Down

Alright, guys, let's get to work! We've got our expression: ${(2)^4 - 3*(2)^3 + 7*(2)^2 - 12}$. Let's break it down step by step so we don't mess anything up.

1. Exponents First: Remember, according to the order of operations (PEMDAS/BODMAS), we tackle exponents before anything else. So, let's calculate the powers: ${2^4 = 16}$, ${2^3 = 8}$, and ${2^2 = 4}$. Our expression now looks like this: ${16 - 3*8 + 7*4 - 12}$.
2. Multiplication: Next up, multiplication! We have two multiplication operations to take care of: ${3*8 = 24}$ and ${7*4 = 28}$. Our expression becomes: ${16 - 24 + 28 - 12}$.
3. Addition and Subtraction: Finally, we do addition and subtraction from left to right. So, ${16 - 24 = -8}$. Then, ${-8 + 28 = 20}$. And lastly, ${20 - 12 = 8}$.

There you have it! We've worked our way through the expression step-by-step, and we've arrived at our answer. Careful execution of these operations is crucial, ensuring no errors are made along the way. Each calculation builds upon the last, and following these steps will lead to a correct outcome every time. Remember, take your time and double-check your work as you go. A systematic approach is your best friend in math. That's the key here, breaking it down into smaller, manageable parts. Each part contributes to the ultimate solution, making the problem far less intimidating than it might first appear. Practicing these steps will help solidify your understanding and boost your confidence when facing similar problems.

The Answer: What We Found

So, after all that number crunching, we found out that the value of ${y^4 - 3y^3 + 7y^2 - 12}$ when ${y = 2}$ is 8. That's the final answer, and we got there by carefully substituting the value of ${y}$ and following the order of operations.

It's like following a recipe: you put in the ingredients (the value of ${y}$), and you follow the instructions (the order of operations), and out pops the result (the value of the expression). See? Not so hard, right?

This value, 8, is the result of our careful substitutions and calculations. This also tells us something cool; for this specific function, at this particular point (y=2), the value of the function is 8. That is just one point on the line represented by the function, and the result is a single coordinate on the function's curve. Understanding this can unlock more advanced topics in mathematics, and helps you grasp how functions behave. It shows that the value depends on the variable, ${y}$, and how changes in ${y}$ affect the function's overall output. This concept is foundational to understanding the relationships between variables and mathematical expressions in more sophisticated settings.

Why This Matters: The Big Picture

This might seem like a small problem, but it's actually super important. This process, which involves substituting a variable's value and evaluating the expression, is the basis for solving many more complex math problems. We use this all the time in algebra, calculus, and even in computer programming. It helps us analyze equations, model real-world situations, and make predictions. Once you understand this concept, you will discover that this approach is useful across a broad range of mathematical and scientific disciplines.

It also teaches you to pay attention to details. When we're working through these problems, every step matters. A small mistake in the order of operations or a calculation error can throw off the entire answer. So, it's all about precision and taking your time. This attention to detail will help you in all your studies, as well as in life. This also has the added benefit of developing important skills that can apply to real-world situations. It's a foundation for analytical thought. The accuracy we strive for in mathematics is applicable to many fields. Whether it's understanding financial models, designing structures, or processing information, a solid grasp of mathematical principles and methods is invaluable.

Tips for Success: Level Up Your Skills

Want to get even better at these types of problems? Here are a few tips:

• Practice Makes Perfect: The more you do these types of problems, the easier they become. Try working through more examples on your own. You can find tons of practice problems online or in textbooks.
• Master the Order of Operations: Make sure you know the order of operations (PEMDAS/BODMAS) inside and out. It's the most important thing to remember.
• Show Your Work: Write out every step, just like we did here. This helps you avoid mistakes and makes it easier to catch errors if you make them. It's a super effective way to learn.
• Double-Check Your Calculations: Don't be afraid to go back and check your work, especially the calculations. Mistakes happen, but that's okay! The key is to learn from them.

By consistently practicing and reinforcing your knowledge, you'll gain the skills and confidence to confidently tackle these challenges. This proactive approach not only enhances your current problem-solving abilities, but also establishes a solid foundation for more advanced mathematical studies.

Conclusion: You've Got This!

And that's it, guys! We've successfully solved the problem! Remember, evaluating an expression at a certain point is a crucial skill, and with practice, you'll become a pro at it. Keep practicing, keep learning, and don't be afraid to ask for help if you need it. You've got this!

So, to recap: We first substituted the given value, then systematically simplified the expression using the order of operations, and arrived at our final answer. This systematic approach is essential in math and science. By breaking down problems into manageable steps, we can reduce complexity and boost our understanding, making even the most daunting tasks easier to manage.

Great job today everyone! Now go out there and keep learning. See ya next time!