Simplify $(7x-1)(3x-4)$: Standard Form Guide

Hey guys! Today, we're diving into a common algebra problem: simplifying the expression $(7x - 1)(3x - 4)$ and writing the result in standard form. Don't worry, it's not as scary as it looks! We'll break it down step by step, so you can confidently tackle similar problems in the future. Let's get started and make math a little less mysterious, and a lot more fun. We will make sure to understand the steps clearly and apply them effectively. So, grab your pencils and let's get to work!

Understanding Standard Form

Before we jump into the simplification, let's quickly recap what standard form actually means for a polynomial. Think of it as the most organized way to present your answer. In standard form, a polynomial is written with the terms arranged in descending order of their exponents. What this means, guys, is that the term with the highest power of the variable comes first, followed by the term with the next highest power, and so on, until we reach the constant term (the one without any variables).

Why is this important? Well, standard form makes it super easy to compare polynomials, identify their degree (the highest power of the variable), and perform further operations like addition or subtraction. It's like having a universal language for polynomials, so everyone knows exactly what you're talking about. For a quadratic expression (which is what we'll end up with here), the standard form looks like this: $ax^2 + bx + c$, where 'a', 'b', and 'c' are constants.

Now, when we talk about descending order of exponents, we're simply saying that we want to arrange the terms from the highest power of $x$ down to the lowest. For instance, if we have an expression like $5x + 3x^2 + 2$, to express it in standard form, we'd rearrange it to $3x^2 + 5x + 2$. Notice how the $x^2$ term comes first, followed by the $x$ term, and then the constant. This systematic approach not only makes our expressions neater but also makes them easier to work with and understand.

So, keep this in mind as we move forward. Our goal isn't just to simplify the expression, but to present our final answer in this clean, organized standard form. It's like making sure your room is not only clean but also well-organized, making it easier to find things later. Let's keep this in mind as we tackle the next steps in our simplification journey!

Step-by-Step Simplification Using the Distributive Property (FOIL)

Okay, guys, let's get down to the nitty-gritty of simplifying the expression $(7x - 1)(3x - 4)$. The key here is the distributive property, which some of you might know by the acronym FOIL: First, Outer, Inner, Last. This handy method helps us make sure we multiply each term in the first set of parentheses by each term in the second set. Think of it as making sure everyone at the party gets a handshake!

Let's break it down:

1. First: Multiply the first terms in each parenthesis: $(7x) * (3x) = 21x^2$
2. Outer: Multiply the outer terms: $(7x) * (-4) = -28x$
3. Inner: Multiply the inner terms: $(-1) * (3x) = -3x$
4. Last: Multiply the last terms: $(-1) * (-4) = 4$

So, after applying the distributive property, we get: $21x^2 - 28x - 3x + 4$. But we're not done yet! We've expanded the expression, but now we need to combine like terms to simplify it further. Combining like terms is like sorting your socks after laundry – you group the ones that are similar. In our case, the like terms are the ones with the same variable and exponent. We have two terms with 'x' to the power of 1: $-28x$ and $-3x$.

Combining these, we get $-28x - 3x = -31x$. Now, we can rewrite our expression as: $21x^2 - 31x + 4$. Look at that! We've simplified the expression by using the distributive property and combining like terms. It's like turning a tangled mess into a neat and orderly result. This is a crucial step in algebra, and mastering it will make your life so much easier when dealing with more complex problems. Remember, FOIL is your friend, and combining like terms is the secret to a tidy solution. So, let's keep this momentum going as we move on to the next step: writing our simplified expression in standard form.

Expressing the Simplified Expression in Standard Form

Alright, we've done the hard work of expanding and simplifying our expression. Now comes the final touch: expressing it in standard form. Remember from our earlier discussion, standard form means arranging the terms in descending order of their exponents. It's like lining up your books on a shelf from the tallest to the shortest – a neat and organized way to present things.

We ended up with the simplified expression $21x^2 - 31x + 4$. Let's take a close look at the exponents of our variable, 'x'. We have a term with $x^2$ (which means x to the power of 2), a term with $x$ (which is understood as x to the power of 1), and a constant term (which can be thought of as x to the power of 0, since any number to the power of 0 is 1).

So, to put this in standard form, we simply need to make sure the term with the highest exponent comes first, followed by the term with the next highest exponent, and so on. In our case, the order is already correct! $21x^2$ is the term with the highest exponent (2), followed by $-31x$ (exponent of 1), and then the constant term $+4$ (exponent of 0).

Therefore, the expression $21x^2 - 31x + 4$ is already in standard form. How cool is that? Sometimes, the math gods are smiling upon us! This might seem like a small step, but it's super important. Presenting your answer in standard form shows that you understand the conventions of algebra and can communicate your results clearly. It also makes it easier for others to understand your work and for you to use the expression in further calculations.

So, there you have it! We've successfully simplified the expression and written it in standard form. It's like adding the perfect frame to a beautiful picture, completing the whole package. Now, let's take a moment to reflect on the journey we've taken and the key takeaways from this exercise.

Final Answer and Conclusion

Okay, guys, let's bring it all together! We started with the expression $(7x - 1)(3x - 4)$, and after a little algebraic maneuvering, we've arrived at our final answer in standard form. Drumroll, please… The simplified expression in standard form is $21x^2 - 31x + 4$.

We've covered some serious ground here, from applying the distributive property (FOIL) to combining like terms and finally arranging our answer in standard form. Each step was like a piece of the puzzle, and now we've put them all together to see the complete picture. Remember, the journey through each step is just as important as the final destination. Understanding the process helps you tackle similar problems with confidence and maybe even a little bit of swagger!

So, what are the key takeaways from this exercise?

• Distributive Property (FOIL): This is your go-to tool for expanding expressions with parentheses. Think First, Outer, Inner, Last, and you'll be multiplying like a pro.
• Combining Like Terms: Don't forget to group similar terms together to simplify your expression. It's like decluttering your math!
• Standard Form: Present your final answer in descending order of exponents. It's the polite thing to do in the math world, and it makes your work easier to understand.

These skills aren't just for this specific problem; they're fundamental to algebra and will come up again and again in your mathematical adventures. It's like learning the basic chords on a guitar – once you've got them down, you can play a whole bunch of songs.

In conclusion, simplifying expressions and writing them in standard form is a core skill in algebra. By mastering these techniques, you're not just solving problems; you're building a solid foundation for future mathematical success. So, keep practicing, keep exploring, and remember, math can be fun! Now, go forth and conquer those algebraic challenges! You've got this!