Simplifying Algebraic Expressions: A Step-by-Step Guide

Hey guys! Ever stumbled upon an algebraic expression that looks like a jumbled mess? Don't worry, we've all been there. Today, we're going to break down a common type of problem: simplifying expressions. Specifically, we'll tackle an expression that looks like this: $5(2x^2 - 1) + 3(7x^2 + 1)$. By the end of this guide, you'll be a pro at simplifying similar expressions. Let's dive in!

Understanding the Expression

Before we jump into the solution, let's first understand what the expression $5(2x^2 - 1) + 3(7x^2 + 1)$ actually means. This expression combines numbers, variables (in this case, 'x'), and mathematical operations. Our goal is to simplify it, which means to rewrite it in a more compact and manageable form.

The key here is to follow the order of operations and apply the distributive property. Remember those good old rules from math class? They're about to become your best friends! We'll be distributing the numbers outside the parentheses into the terms inside, and then we'll combine like terms. Think of it like decluttering – we're taking a messy expression and organizing it into something neat and tidy.

Why is this important? Well, simplified expressions are much easier to work with. Whether you're solving equations, graphing functions, or just trying to understand a mathematical concept, having a simplified expression makes the process smoother and less prone to errors. So, let's get started on making this expression shine!

Step-by-Step Solution

Okay, let's get down to business and simplify the expression $5(2x^2 - 1) + 3(7x^2 + 1)$ step by step. We'll break it down into manageable chunks so it's super easy to follow.

Step 1: Apply the Distributive Property

The first thing we need to do is get rid of those parentheses. To do that, we'll use the distributive property. Remember, this means we multiply the number outside the parentheses by each term inside.

So, for the first part, $5(2x^2 - 1)$, we multiply 5 by both $2x^2$ and -1:

• $5 * 2x^2 = 10x^2$
• $5 * -1 = -5$

That gives us $10x^2 - 5$.

Now, let's do the same for the second part, $3(7x^2 + 1)$. We multiply 3 by both $7x^2$ and 1:

• $3 * 7x^2 = 21x^2$
• $3 * 1 = 3$

That gives us $21x^2 + 3$.

Step 2: Combine Like Terms

Now that we've distributed, our expression looks like this: $10x^2 - 5 + 21x^2 + 3$. The next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, our like terms are the $x^2$ terms and the constant terms.

Let's group them together:

• $x^2$ terms: $10x^2 + 21x^2$
• Constant terms: $-5 + 3$

Now, let's add them up:

• $10x^2 + 21x^2 = 31x^2$
• $-5 + 3 = -2$

Step 3: Write the Simplified Expression

We've done the hard work! Now we just need to put it all together. We have $31x^2$ and -2. So, our simplified expression is:

$31x^2 - 2$

And that's it! We've successfully simplified the expression. Wasn't so bad, right? The key is to take it one step at a time and remember those basic rules of algebra.

Why This Works: A Deeper Dive

So, we've simplified the expression, but let's take a moment to understand why these steps work. Knowing the underlying principles can make you a more confident problem-solver.

The Distributive Property: Sharing is Caring (Numbers, That Is)

The distributive property is the foundation of our first step. It basically says that multiplying a number by a group of numbers added together is the same as multiplying the number by each individual number in the group and then adding the results. In mathematical terms:

$a(b + c) = ab + ac$

Think of 'a' as the shareholder and '(b + c)' as the company. 'a' needs to "share" its multiplication with both 'b' and 'c'.

In our case, we used it like this:

• $5(2x^2 - 1) = (5 * 2x^2) + (5 * -1)$
• $3(7x^2 + 1) = (3 * 7x^2) + (3 * 1)$

This property allows us to break down the expression and get rid of those pesky parentheses.

Combining Like Terms: Finding Your Group

Combining like terms is like organizing your closet. You put all the shirts together, all the pants together, and so on. In algebra, like terms are terms that have the same variable raised to the same power. For example, $3x^2$ and $7x^2$ are like terms because they both have 'x' raised to the power of 2. However, $3x^2$ and $7x$ are not like terms because the powers of 'x' are different.

We can only add or subtract like terms. This is because they represent the same "thing." Think of $3x^2$ as 3 squares and $7x^2$ as 7 squares. It makes sense to combine them into 10 squares ($10x^2$). But you can't directly add squares and single 'x's because they're different shapes, so to speak.

Order of Operations: The Rules of the Game

Underlying all of this is the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This is the set of rules that dictates the order in which we perform operations. We distribute first (which is a form of multiplication), and then we combine like terms (which involves addition and subtraction).

By following these rules, we ensure that we simplify the expression correctly and arrive at the right answer.

Common Mistakes to Avoid

Alright, guys, let's talk about some common pitfalls that people often stumble into when simplifying expressions. Knowing these mistakes can help you avoid them and ace your algebra problems!

Forgetting the Distributive Property

The biggest mistake is forgetting to distribute the number outside the parentheses to every term inside. It's super tempting to just multiply it by the first term and move on, but that's a recipe for disaster! Remember, that number is like a gift that needs to be shared with everyone inside the parentheses.

Example: If you have $3(x + 2)$, you need to multiply 3 by both 'x' and 2, giving you $3x + 6$. Don't just write $3x + 2$! That's a big no-no.

Mixing Up Like Terms

Another common mistake is combining terms that aren't actually "like." Remember, like terms have the same variable and the same exponent. You can't add $x^2$ and $x$ together, just like you can't add apples and oranges.

Example: $5x^2 + 2x$ cannot be simplified further. They're different terms! You can't combine them into $7x^3$ or $7x^2$ or anything like that.

Sign Errors

Watch out for those sneaky negative signs! They can easily trip you up. When distributing, pay close attention to the signs of the terms inside the parentheses.

Example: In the expression $2(x - 3)$, you're actually multiplying 2 by -3, so the result is $-6$, not 6. So, the correct expansion is $2x - 6$.

Order of Operations Faux Pas

Sometimes, people get a bit confused about the order of operations and try to add or subtract terms before distributing. Remember PEMDAS! Distribution (which is a form of multiplication) comes before addition and subtraction.

Example: In the expression $4 + 2(x + 1)$, you need to distribute the 2 first, then add the 4. So, it's $4 + 2x + 2$, which simplifies to $2x + 6$. Don't add the 4 and 2 first!

Careless Arithmetic

Sometimes, the mistake isn't even about algebra – it's just a simple arithmetic error. Make sure you're adding, subtracting, multiplying, and dividing correctly, especially when dealing with negative numbers.

Tip: If you're prone to arithmetic errors, use a calculator to double-check your work, especially on more complex problems.

By being aware of these common mistakes, you can significantly improve your accuracy when simplifying expressions. So, keep these in mind, double-check your work, and you'll be simplifying like a pro in no time!

Practice Problems

Okay, now that we've gone through the steps and talked about common mistakes, it's time to put your knowledge to the test! Practice makes perfect, so let's work through a few more examples together.

Problem 1: Simplify $2(3x^2 + 4) - (x^2 - 1)$

• Step 1: Distribute.
  • $2 * 3x^2 = 6x^2$
  • $2 * 4 = 8$
  • $-1 * x^2 = -x^2$ (Remember, the negative sign in front of the parentheses means we're distributing a -1)
  • $-1 * -1 = 1$
  • This gives us: $6x^2 + 8 - x^2 + 1$
• Step 2: Combine like terms.
  • $x^2$ terms: $6x^2 - x^2 = 5x^2$
  • Constant terms: $8 + 1 = 9$
• Step 3: Write the simplified expression.
  • $5x^2 + 9$

Problem 2: Simplify $-4(2x - 3) + 5(x + 2)$

• Step 1: Distribute.
  • $-4 * 2x = -8x$
  • $-4 * -3 = 12$ (Careful with the signs!)
  • $5 * x = 5x$
  • $5 * 2 = 10$
  • This gives us: $-8x + 12 + 5x + 10$
• Step 2: Combine like terms.
  • $x$ terms: $-8x + 5x = -3x$
  • Constant terms: $12 + 10 = 22$
• Step 3: Write the simplified expression.
  • $-3x + 22$

Problem 3: Simplify $7x^2 - 3(x^2 + 2) + 4$

• Step 1: Distribute.
  • $-3 * x^2 = -3x^2$
  • $-3 * 2 = -6$
  • This gives us: $7x^2 - 3x^2 - 6 + 4$
• Step 2: Combine like terms.
  • $x^2$ terms: $7x^2 - 3x^2 = 4x^2$
  • Constant terms: $-6 + 4 = -2$
• Step 3: Write the simplified expression.
  • $4x^2 - 2$

See how it's done? Take it one step at a time, be mindful of the signs, and don't forget to distribute! The more you practice, the easier it will become.

Conclusion

And there you have it, folks! We've successfully simplified the expression $5(2x^2 - 1) + 3(7x^2 + 1)$ and learned the ins and outs of simplifying algebraic expressions in general. Remember, the key is to take it step by step: distribute, combine like terms, and watch out for those common mistakes.

Simplifying expressions is a fundamental skill in algebra, and mastering it will make your mathematical journey much smoother. So, keep practicing, stay confident, and you'll be tackling even the trickiest expressions with ease. You got this! 🚀

The simplified form of the expression $5(2x^2 - 1) + 3(7x^2 + 1)$ is $31x^2 - 2$. So, the correct answer is D. Great job if you got it right!