Simplify -15p² + 7p + P + 4p² - 2p³: Step-by-Step Solution

Hey guys! Today, we're diving into the world of polynomials and tackling the expression -15p² + 7p + p + 4p² - 2p³. Don't worry if it looks intimidating at first; we'll break it down step-by-step. Think of it like decluttering – we're going to group similar items together to make things simpler and easier to understand. Polynomials are algebraic expressions that consist of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. They are fundamental in algebra and calculus, forming the building blocks for more complex mathematical concepts. Understanding how to simplify polynomials is crucial for solving equations, graphing functions, and many other mathematical tasks. Polynomials appear in various real-world applications, from engineering and physics to economics and computer science. For instance, they can model the trajectory of a projectile, the growth of a population, or the behavior of financial markets. This versatility makes mastering polynomial manipulation an essential skill for anyone pursuing a career in STEM fields. When simplifying polynomials, the main goal is to combine like terms, which are terms that have the same variable raised to the same power. This process involves adding or subtracting the coefficients of these like terms while keeping the variable and exponent the same. For example, in the expression 3x² + 2x - x² + 5x, the like terms are 3x² and -x², as well as 2x and 5x. Combining these terms gives us (3 - 1)x² + (2 + 5)x, which simplifies to 2x² + 7x. The simplified form of a polynomial is generally written in descending order of exponents, meaning the term with the highest exponent comes first, followed by terms with lower exponents. This standard form makes it easier to compare and work with different polynomials. So, grab your thinking caps, and let's get started on simplifying this expression!

Understanding the Expression: -15p² + 7p + p + 4p² - 2p³

Alright, let's get a good look at our expression: -15p² + 7p + p + 4p² - 2p³. The first thing we need to do is identify the different terms. Think of terms as individual pieces of the expression, separated by plus or minus signs. In this case, we have five terms: -15p², 7p, p, 4p², and -2p³. Each term consists of a coefficient (the number part) and a variable part (the 'p' raised to some power). For instance, in the term -15p², -15 is the coefficient, and p² is the variable part. The degree of a term is the exponent of the variable. So, -15p² has a degree of 2, 7p has a degree of 1 (since p is the same as p¹), and -2p³ has a degree of 3. Understanding the degree of a term is crucial for organizing and simplifying polynomials. Now, let's talk about like terms. Like terms are terms that have the same variable raised to the same power. They're like the apples and oranges in our algebraic fruit basket – we can only combine apples with apples and oranges with oranges. Looking at our expression, we can see that -15p² and 4p² are like terms because they both have p raised to the power of 2. Similarly, 7p and p are like terms because they both have p raised to the power of 1. The term -2p³ is unique because it's the only term with p raised to the power of 3. Identifying like terms is the key to simplifying polynomials. Once we know which terms are alike, we can combine them by adding or subtracting their coefficients. This is similar to combining like fractions, where you can only add or subtract fractions with the same denominator. For example, just as you can add 1/4 and 2/4 to get 3/4, you can add 2x and 3x to get 5x. So, with our terms identified and like terms recognized, we're ready to start simplifying! The next step is to rearrange the terms to group the like terms together. This will make it easier to see which terms can be combined and avoid any mistakes. Think of it like organizing your closet – grouping similar items together makes everything more manageable. Once we've grouped the like terms, we can move on to the actual process of combining them. Stay tuned, because that's where the real simplification magic happens!

Step 1: Rearranging the Terms

Okay, guys, now that we've identified our terms and know which ones are like terms, let's get organized! The first step in simplifying our expression, -15p² + 7p + p + 4p² - 2p³, is to rearrange the terms so that the like terms are next to each other. This is like sorting your laundry – you put all the socks together, all the shirts together, and so on. In math, we're putting all the p² terms together, all the p terms together, and any constant terms (if we had any) together. But before we just shuffle things around, let's also think about the order we want to present our final simplified expression. Mathematicians usually like to write polynomials in descending order of exponents. This means we put the term with the highest power of the variable first, then the term with the next highest power, and so on, until we get to the constant term (which has a power of 0). It's like lining up students by height, from tallest to shortest. So, in our expression, the term with the highest power is -2p³, which has p raised to the power of 3. That's going to be our first term. Next, we look for terms with p raised to the power of 2. We have two of those: -15p² and 4p². These will come next. Then, we have terms with p raised to the power of 1 (which we just write as p). We have 7p and p. These will follow. And finally, if we had any constant terms (numbers without any p's), they would go last. With this in mind, let's rearrange our expression. We start with -2p³, then we bring together -15p² and 4p², and finally, we bring together 7p and p. This gives us: -2p³ - 15p² + 4p² + 7p + p See how we've grouped the like terms together? Now it's much easier to see which terms we can combine. Rearranging terms doesn't change the value of the expression, just like rearranging furniture in a room doesn't change the size of the room. We're just making it easier to work with. This step is super important because it sets us up for the next step: combining like terms. When the like terms are grouped together, we can focus on their coefficients and perform the addition or subtraction more easily. It's like having all your ingredients prepped before you start cooking – it makes the whole process smoother and less likely to result in mistakes. So, with our terms neatly rearranged, we're ready to move on to the next step and do some combining!

Step 2: Combining Like Terms

Alright, rockstars, we've got our expression neatly rearranged as -2p³ - 15p² + 4p² + 7p + p. Now comes the fun part: combining those like terms! Remember, like terms are those that have the same variable raised to the same power. We've already grouped them together, so this should be a breeze. Think of it like counting apples and oranges separately. You wouldn't say you have