Polynomial Difference: Solving $\left(7x^2+8\right)-\left(4x^2+x+6\right)$

Hey guys! Let's dive into a common algebra problem: finding the polynomial that represents the difference between two given polynomials. Specifically, we're tackling $\left(7x^2+8\right)-\left(4x^2+x+6\right)$. Don't worry, it's not as scary as it looks. We'll break it down step-by-step to make sure everyone understands. This is a fundamental concept in algebra, so grasping this will help you tackle more complex problems later on. We'll go through the process of simplifying the expression and identifying the correct answer from the provided choices. Ready to get started?

Understanding the Problem: Polynomial Subtraction

First things first, what exactly does it mean to subtract polynomials? Well, it's similar to subtracting regular numbers, but we have to keep track of the variables and their exponents. When we subtract, we're essentially finding the difference between the terms of the two polynomials. The key here is to remember that we're subtracting the entire second polynomial, which means we need to distribute that negative sign. Think of it like this: the minus sign in front of the parentheses changes the sign of each term inside. So, our expression $\left(7x^2+8\right)-\left(4x^2+x+6\right)$ transforms into something a bit more manageable once we handle the subtraction properly. It's really all about paying close attention to those minus signs; they're the sneaky culprits that can trip you up if you're not careful. The goal is to combine like terms after distributing the negative sign, which simplifies the expression to a single polynomial. Let's get our hands dirty and figure out how to solve this.

Now, let's break down the given problem: $\left(7x^2+8\right)-\left(4x^2+x+6\right)$. We need to find the polynomial that results from this subtraction. The options provided are:

A. $11x^2+x+14$

B. $3x^2-x+2$

C. $11x^2-x+2$

D. $3x^2+x+14$

Our task is to correctly identify which of these polynomials matches the result of our subtraction.

Step-by-Step Solution: Subtracting Polynomials

Alright, let's roll up our sleeves and get to work! The first thing we need to do is distribute that negative sign. Remember, the negative sign in front of the second set of parentheses changes the sign of each term inside. So, we rewrite the expression as follows: $7x^2 + 8 - 4x^2 - x - 6$. See how the signs of $4x^2$, $x$, and $6$ have flipped? This is a super crucial step! It’s also important to be organized. This will prevent mistakes in more complex problems. Now, the fun part begins: combining like terms! Like terms are terms that have the same variable raised to the same power. So, $7x^2$ and $-4x^2$ are like terms, as are the constants $8$ and $-6$. We combine them separately. We combine the $x^2$ terms: $7x^2 - 4x^2 = 3x^2$. Then, we have the $-x$ term, which doesn't have any other like terms, so it just stays as is. Finally, we combine the constant terms: $8 - 6 = 2$. Putting it all together, we get $3x^2 - x + 2$. See, that wasn't so bad, right? We've successfully simplified the expression, and now we can easily compare it to the answer choices provided.

Let's meticulously go through the steps again.

1. Distribute the Negative Sign: This means changing the sign of each term inside the second parentheses. Our original expression $\left(7x^2+8\right)-\left(4x^2+x+6\right)$ becomes $7x^2 + 8 - 4x^2 - x - 6$.

2. Combine Like Terms: Identify and combine terms with the same variable and exponent. Here, we combine $x^2$ terms: $7x^2 - 4x^2 = 3x^2$. We also combine the constant terms: $8 - 6 = 2$. The $-x$ term remains unchanged.

3. Write the Simplified Polynomial: Combine all the results from the previous step to get the final simplified polynomial. This gives us $3x^2 - x + 2$.

Identifying the Correct Answer

Now that we've crunched the numbers and simplified our expression to $3x^2 - x + 2$, we can compare it to the answer choices. Let’s look back at the options:

A. $11x^2+x+14$

B. $3x^2-x+2$

C. $11x^2-x+2$

D. $3x^2+x+14$

Looking at the options, we can see that answer B, $3x^2 - x + 2$, is the correct one. It matches the simplified polynomial we calculated. The other options have incorrect coefficients or signs, which means they are not equivalent to the original expression. Therefore, by carefully distributing the negative sign and combining like terms, we’ve successfully found the right answer. We can confidently select option B as the correct solution. It's a great feeling, isn't it? Getting to the correct answer step by step. This method ensures that we can solve similar problems confidently in the future. Remember to keep practicing and always double-check your work!

Common Mistakes and How to Avoid Them

Even the best of us make mistakes! Let's talk about some common pitfalls when subtracting polynomials and how to steer clear of them. One frequent error is forgetting to distribute the negative sign to every term inside the second set of parentheses. This can lead to incorrect signs on the terms and, consequently, the wrong answer. Always remember to change the sign of each term! Another common mistake is combining unlike terms. Make sure you only combine terms that have the same variable and exponent. For example, you can't add an $x^2$ term and an $x$ term. Keep those terms separate!

To avoid these mistakes, here are a few handy tips:

• Write it out: Always rewrite the expression with the negative sign distributed to make sure you're clear on the signs of each term.
• Highlight like terms: Use different colors or underlines to group like terms. This can make it easier to see which terms you need to combine.
• Double-check your work: After you simplify, go back and re-evaluate each step. It’s always good to make sure you didn’t make any simple math errors!

By keeping these common mistakes in mind and following these tips, you'll be well on your way to mastering polynomial subtraction. Consistency and practice are your best friends here. Good luck! Keep up the good work, and you will become experts at simplifying these expressions in no time!

Conclusion: Mastering Polynomial Subtraction

So, there you have it! We've successfully navigated the world of polynomial subtraction. We've taken a look at the original problem, broken it down step by step, and found the correct answer. Remember the key steps: distributing the negative sign, combining like terms, and staying organized. Polynomial subtraction is a fundamental concept in algebra, so understanding it will help you in future math adventures. It's a valuable skill that opens doors to more complex problems. Remember that practice is key, and with each problem you solve, you'll become more confident and proficient. Keep practicing, and you'll be acing these problems in no time! Keep up the great work, and I hope this helped you understand polynomial subtraction better!