Simplify (7x^2 - 6x + 2) - (4x - 8) + (-6x^2 + 3x)

Hey everyone! In this article, we're going to tackle a common algebra problem: simplifying polynomial expressions. Polynomials might sound intimidating, but they're really just expressions with variables and numbers combined using addition, subtraction, and multiplication. Today, we'll break down the steps to simplify the expression $(7x^2 - 6x + 2) - (4x - 8) + (-6x^2 + 3x)$. So, grab your pencils and let's dive in!

Understanding Polynomials

Before we jump into the problem, let's quickly review what polynomials are. A polynomial is an expression consisting of variables (like x) and coefficients (numbers) combined using mathematical operations such as addition, subtraction, and multiplication. The exponents of the variables must be non-negative integers.

Think of polynomials as mathematical LEGOs. Each term, like $7x^2$ or $-6x$, is a LEGO brick. The entire polynomial is the structure you build by combining these bricks. Understanding this basic concept is crucial for simplifying complex expressions.

For example: $3x^2 + 2x - 1$ is a polynomial. $5x^3 - x + 4$ is also a polynomial. But $2x^{-1} + 1$ is not a polynomial because it has a negative exponent.

Key components of a polynomial term:

• Coefficient: The numerical part of the term (e.g., 7 in $7x^2$).
• Variable: The symbol representing an unknown value (e.g., x).
• Exponent: The power to which the variable is raised (e.g., 2 in $x^2$).

Simplifying polynomials involves combining like terms. Like terms are terms that have the same variable raised to the same power. For instance, $7x^2$ and $-6x^2$ are like terms, but $7x^2$ and $-6x$ are not. Combining like terms is like grouping similar LEGO bricks together to make the structure simpler and more organized. This is a fundamental concept that we will be using throughout this simplification process. We need to identify and then combine these like terms, which will involve both addition and subtraction, ensuring we follow the correct order of operations. Remember, the goal is to make the expression as concise and understandable as possible.

Step 1: Distribute the Negative Sign

Okay, let's get started with our expression: $(7x^2 - 6x + 2) - (4x - 8) + (-6x^2 + 3x)$. The first thing we need to do is deal with the subtraction. Remember, subtracting a group of terms is the same as adding the negative of each term. So, we need to distribute the negative sign in front of the parentheses $(4x - 8)$.

Think of it like this: the minus sign acts like a multiplier of -1. We're essentially multiplying each term inside the parentheses by -1. This is a critical step because it ensures we handle the signs correctly. Forgetting to distribute the negative sign is a common mistake, and it can lead to an incorrect answer.

So, let's rewrite the expression:

$(7x^2 - 6x + 2) - (4x - 8) + (-6x^2 + 3x)$ becomes

$7x^2 - 6x + 2 - 4x + 8 + (-6x^2 + 3x)$.

Notice how the $-4x$ becomes $-4x$ and the $-(-8)$ becomes $+8$. Distributing the negative sign is like flipping the signs of all the terms inside the parentheses. This step is crucial for accurate simplification. Now, our expression is ready for the next step: combining like terms. We've removed the parentheses that were causing the subtraction issue, and we've correctly accounted for the change in signs. This sets us up perfectly to bring together the terms that are similar and simplify the overall expression.

Step 2: Remove Parentheses

Now that we've distributed the negative sign, we can remove the parentheses. This step is pretty straightforward. We're just rewriting the expression without the parentheses, making it easier to see and group the like terms. Think of it as decluttering your workspace before you start organizing your tools. Removing parentheses makes the terms more visible and less cluttered, which helps in identifying and combining like terms accurately.

Our expression now looks like this:

$7x^2 - 6x + 2 - 4x + 8 - 6x^2 + 3x$

Notice that we simply dropped the parentheses around the first group $(7x^2 - 6x + 2)$ and the last group $(-6x^2 + 3x)$. This is because there was no operation directly affecting these groups from the outside (other than the implied positive sign). This step is essential for clarity. With the parentheses gone, we can clearly see all the individual terms and their signs. It's like having a clear roadmap for the rest of the simplification process. We've set the stage perfectly for the next crucial step: identifying and combining the like terms.

Step 3: Identify Like Terms

This is where the real fun begins! Remember, like terms are terms that have the same variable raised to the same power. It's like sorting your LEGO bricks by shape and size. We need to group together the terms with $x^2$, the terms with $x$, and the constant terms (the numbers without any variables).

Let's rewrite our expression and highlight the like terms:

$7x^2$ - 6x + 2 - 6x^2 - 4x + 8 + 3x

Here, we've bolded the $x^2$ terms and italicized the $x$ terms. The constant terms (2 and 8) are left as they are. This visual aid helps us keep track of which terms belong together. This step is fundamental to simplifying polynomials. Identifying like terms correctly ensures that we are combining only those terms that can be combined, maintaining the integrity of the expression. It's like making sure you're adding apples to apples and not apples to oranges. The process of identifying like terms involves carefully examining each term and comparing its variable part (including the exponent) with that of other terms. Once we've correctly identified the like terms, we can move on to the final step: combining them.

Step 4: Combine Like Terms

Now for the grand finale! We're going to combine the like terms we identified in the previous step. This is like adding up all the bricks of the same shape to see how many you have in total. Remember, when combining like terms, we only add or subtract the coefficients (the numbers in front of the variables). The variable and its exponent stay the same.

Let's start with the $x^2$ terms: $7x^2 - 6x^2$. This is like saying we have 7 of something and we take away 6. The result is:

$7x^2 - 6x^2 = 1x^2$ or simply $x^2$.

Next, let's combine the $x$ terms: $-6x - 4x + 3x$. This is like saying we owe 6 dollars, then we owe 4 more dollars, but then we get 3 dollars. The result is:

$-6x - 4x + 3x = -10x + 3x = -7x$.

Finally, let's combine the constant terms: $2 + 8$. This is a simple addition:

$2 + 8 = 10$.

Now, let's put it all together. We combine the simplified terms to get our final simplified expression. This is the last stage of our simplification journey, where all the pieces come together to form the final answer. This is the most important step as this gives us the reduced form of the polynomial expression.

Final Answer

Putting all the simplified terms together, we get:

$x^2 - 7x + 10$

And there you have it! We've successfully simplified the polynomial expression $(7x^2 - 6x + 2) - (4x - 8) + (-6x^2 + 3x)$ to $x^2 - 7x + 10$. Woohoo! This is the moment of triumph, where all our hard work pays off. We've taken a complex-looking expression and broken it down into its simplest form. This not only makes the expression easier to understand but also easier to work with in future calculations.

Tips for Success

• Double-check your signs: Pay close attention to the signs, especially when distributing the negative sign. A small mistake with a sign can throw off the entire answer.
• Stay organized: Write out each step clearly and neatly. This makes it easier to track your work and spot any errors.
• Practice makes perfect: The more you practice simplifying polynomial expressions, the easier it will become. So, keep at it!

Simplifying polynomial expressions might seem tricky at first, but with practice and a clear understanding of the steps, you'll become a pro in no time. Remember to distribute negative signs carefully, identify like terms accurately, and combine them patiently. You got this! Keep practicing, and you'll master this skill in no time. Simplifying polynomial expressions is a foundational skill in algebra, and mastering it will open doors to more advanced topics. So, embrace the challenge, enjoy the process, and celebrate your successes along the way!