Polynomial Operations Explained Step-by-Step Solution
Hey guys! Today, we're diving into the exciting world of polynomial operations. Polynomials might sound intimidating, but trust me, they're super manageable once you understand the basics. We're going to break down a problem step-by-step, making it crystal clear how to tackle these kinds of questions. So, let's get started!
Understanding Polynomial Basics
Before we jump into the main problem, let's quickly recap what polynomials are. Polynomials are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. Think of them as building blocks of algebra. You've probably seen them before – expressions like 3x³ + 2x² – 5 are classic examples. The key thing to remember is that the exponents on the variables must be whole numbers (0, 1, 2, 3, and so on).
Now, let's talk about the different parts of a polynomial. We have terms, which are the individual parts separated by plus or minus signs (e.g., 3x³, 2x², and -5 in our example). Each term has a coefficient, which is the number multiplying the variable (e.g., 3, 2, and -5). And then there's the degree, which is the highest exponent of the variable in the polynomial (3 in our example). Understanding these basic components is crucial for performing operations on polynomials effectively. When dealing with polynomial equations, it's like playing a mathematical puzzle where each piece (term) has its own role. The coefficients act as multipliers, the variables as placeholders, and the exponents determine the shape of the expression. Getting familiar with these elements allows us to manipulate and simplify polynomials with confidence.
Common Polynomial Operations
The operations we can perform on polynomials are similar to those we do with regular numbers: addition, subtraction, multiplication, and division. However, there are a few special rules to keep in mind. When adding or subtracting polynomials, we can only combine like terms – that is, terms with the same variable and exponent. For example, we can add 3x² and 5x² to get 8x², but we can't directly add 3x² and 5x³ because they have different exponents. Addition and subtraction are all about grouping similar components, making the process neat and organized.
Multiplication involves distributing each term of one polynomial to every term of the other. This might sound complicated, but it's just a matter of being systematic and careful with your calculations. Division, on the other hand, can be a bit trickier and sometimes involves long division, similar to what you might have learned with numbers. Polynomial division helps us break down complex expressions into simpler forms. Each operation has its own set of rules and techniques, but mastering them opens the door to solving more intricate algebraic problems. Think of polynomial operations as a toolkit; the more tools you know how to use, the more complex structures you can build and understand.
Problem Breakdown: Step-by-Step Solution
Alright, let's tackle the problem at hand. We're given three polynomials:
- P(x) = 3x³ + 2x² – 5
- Q(x) = 2x³ - 5x + 7
- R(x) = -5x³ + 2x + 4
And we need to find the result of {2P(x) + Q(x)} - 3R(x). This looks a bit daunting at first, but we'll break it down into manageable steps. The key is to follow the order of operations (PEMDAS/BODMAS) and take it one step at a time. First, we'll handle the multiplications, then the addition, and finally the subtraction. This systematic approach helps avoid errors and keeps the process organized. Remember, patience and precision are your best friends when working with polynomials. Rushing through the steps can lead to mistakes, so let's take our time and get it right!
Step 1: Calculate 2P(x)
The first thing we need to do is multiply the polynomial P(x) by 2. This means we multiply each term in P(x) by 2:
2P(x) = 2 * (3x³ + 2x² – 5) = 6x³ + 4x² – 10
It's a straightforward application of the distributive property, ensuring each term gets its fair share of the multiplication. Multiplying a polynomial by a constant is like scaling it up or down, but the basic structure remains the same. This step is crucial because it sets the stage for the subsequent additions and subtractions. By accurately performing this multiplication, we ensure that the rest of the solution builds on a solid foundation. Keep an eye on the signs – a small mistake here can throw off the entire calculation! So, double-check each term to make sure everything is in order.
Step 2: Calculate 3R(x)
Next, we need to multiply the polynomial R(x) by 3:
3R(x) = 3 * (-5x³ + 2x + 4) = -15x³ + 6x + 12
Again, we distribute the multiplication across each term. Notice the negative sign in front of the 5x³ term – this is important! When multiplying, always pay close attention to the signs. Just like in the previous step, this multiplication prepares us for the next operation, subtraction. A correct multiplication here ensures that our final answer will be accurate. It's a bit like baking; you need the right ingredients in the right proportions to get the perfect cake. So, let's make sure we have all our terms and signs in place!
Step 3: Calculate 2P(x) + Q(x)
Now, we add 2P(x) and Q(x). Remember, we can only add like terms:
2P(x) + Q(x) = (6x³ + 4x² – 10) + (2x³ - 5x + 7)
Combining like terms, we get:
= (6x³ + 2x³) + 4x² - 5x + (-10 + 7)
= 8x³ + 4x² - 5x - 3
Adding polynomials is all about identifying and combining the terms that match. Think of it like sorting socks – you only pair up socks of the same color and size! In this case, we're pairing up terms with the same variable and exponent. The 6x³ and 2x³ combine to give 8x³, the 4x² term stays as it is since there's no other x² term, and so on. The constant terms -10 and 7 combine to give -3. This step is a crucial part of simplifying the expression and getting closer to the final answer. Accuracy in this step ensures that the subtraction we perform next will be based on a solid foundation.
Step 4: Calculate {2P(x) + Q(x)} - 3R(x)
Finally, we subtract 3R(x) from the result we got in the previous step:
{2P(x) + Q(x)} - 3R(x) = (8x³ + 4x² - 5x - 3) - (-15x³ + 6x + 12)
Remember that subtracting a negative is the same as adding, so we distribute the negative sign:
= 8x³ + 4x² - 5x - 3 + 15x³ - 6x - 12
Now, combine like terms:
= (8x³ + 15x³) + 4x² + (-5x - 6x) + (-3 - 12)
= 23x³ + 4x² - 11x - 15
Subtracting polynomials can be a bit tricky because of the negative signs. It's like dealing with debts – you have to be careful to keep track of what you owe! When subtracting a polynomial, we essentially change the signs of all the terms in the polynomial being subtracted and then add. This is why we distributed the negative sign in the step above. Then, it's just a matter of combining like terms, just like we did in the addition step. The 8x³ and 15x³ combine to give 23x³, the -5x and -6x combine to give -11x, and the -3 and -12 combine to give -15. By carefully managing the signs and combining the correct terms, we arrive at the final, simplified polynomial.
The Final Answer and Conclusion
So, the final result of {2P(x) + Q(x)} - 3R(x) is:
23x³ + 4x² - 11x - 15
Therefore, the correct answer is (d). Congratulations, we've successfully navigated through the world of polynomial operations! By breaking down the problem into smaller, manageable steps, we were able to tackle it with confidence. Remember, the key to mastering polynomials is understanding the basic concepts, being organized, and paying close attention to detail. Practice makes perfect, so keep working on these types of problems, and you'll become a polynomial pro in no time!
Polynomial operations are a fundamental part of algebra, and they show up in various areas of mathematics and beyond. From calculus to engineering, understanding how to manipulate polynomials is essential. So, keep practicing, and you'll be well-equipped to tackle any algebraic challenge that comes your way.
