Algebraic Expression N: Simplify & Prove Divisibility By 12

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Algebraic Expression N: Simplify & Prove Divisibility by 12

Hey there, math enthusiasts and problem-solvers! Ever looked at a bunch of numbers, variables, and parentheses and thought, "Whoa, where do I even begin?" Well, you're in the right place, because today we're going to tackle just such a beast: algebraic expression N. We're talking about simplifying it down to a neat, manageable form and then, get this, proving that it's always divisible by a specific number. This isn't just about crunching numbers; it's about building your analytical muscles and seeing the elegance in mathematics. We’ll break down this complex problem into bite-sized, digestible steps, making sure you grasp every single concept along the way. So, grab your imaginary (or real!) calculator, a fresh cup of coffee, and let's dive deep into the fascinating world of algebraic simplification and divisibility proofs.

Unraveling the Mystery: What's N All About?

Alright, guys, let's kick things off by properly introducing our star for today: the algebraic expression N. This expression might look a bit intimidating at first glance, packed with x's and operations, but trust me, it's just a puzzle waiting to be solved. Understanding what expression N represents is our first crucial step. We're given N as: N = -(4x+3) - (4x+3)(4x+2) + (2x+1) - (2x+2)(2x-4). Our mission, should we choose to accept it (and we definitely do!), is twofold. First, for any real number x, we need to show that this hefty expression N can be boiled down to a much simpler form: N = -6(2x+2). Think of it like taking a messy pile of LEGO bricks and assembling them into a sleek, identifiable model. The second part of our challenge is to then demonstrate that, if x is a natural number, this simplified N is always divisible by 12. This isn't just a random exercise; it teaches us invaluable skills in algebraic manipulation, careful distribution, and applying divisibility rules. We'll navigate through common algebraic pitfalls, highlight key strategies, and ensure you feel confident in your ability to tackle similar problems in the future. Remember, every complex problem is just a series of simpler problems strung together. We're going to dissect each component of N, expand it, combine like terms, and work towards our target form. This process isn't just about getting the right answer; it's about understanding the journey of how we get there, building a solid foundation in your mathematical thinking. Pay close attention to the signs, distribute carefully, and remember that patience is a virtue in algebra!

Part A: Proving N = -6(2x+2) – Let's Get Solving!

Now for the fun part, guys! Our first big task is to prove that the intimidating algebraic expression N actually simplifies to N = -6(2x+2) for any real number x. This involves a bit of grunt work, carefully expanding products and combining like terms. It’s like being a detective, looking for clues (terms) and putting them together. We'll start by breaking down the products within N. The full expression is N = -(4x+3) - (4x+3)(4x+2) + (2x+1) - (2x+2)(2x-4). We need to be super meticulous with our multiplications and, most importantly, our signs! A single misplaced negative sign can throw the entire calculation off, so let's approach this with focus and precision. We’ll tackle the two big multiplication terms first, (4x+3)(4x+2) and (2x+2)(2x-4), using the FOIL method (First, Outer, Inner, Last) or simply by distributing each term from the first parenthesis to the second. This method ensures we don't miss any parts of the expansion. Once we have these expanded, we'll substitute them back into the main expression for N, taking great care with the negative signs preceding some of these products. After that, it's a matter of identifying and combining all the x^2 terms, x terms, and constant terms. This process, while seemingly tedious, is absolutely fundamental to mastering algebra. It strengthens your ability to see patterns, manage multiple operations, and simplify complex structures into elegant forms. Don't rush through this section; the devil, or in this case, the solution, is truly in the details. We'll walk through each step, making sure you're comfortable before moving on to the final combination.

Breaking Down the First Product: (4x+3)(4x+2)

Let's zero in on the first product we need to expand: (4x+3)(4x+2). This is a classic example where the FOIL method really shines. Remember, FOIL stands for First, Outer, Inner, Last, and it helps ensure you multiply every term by every other term exactly once. So, let's apply it carefully:

  • First: Multiply the first terms in each parenthesis: (4x) * (4x) = 16x^2.
  • Outer: Multiply the outer terms: (4x) * (2) = 8x.
  • Inner: Multiply the inner terms: (3) * (4x) = 12x.
  • Last: Multiply the last terms in each parenthesis: (3) * (2) = 6.

Now, we combine these results: 16x^2 + 8x + 12x + 6. Simplifying the x terms, we get: 16x^2 + 20x + 6. Easy peasy, right? This expanded form will be substituted back into our main expression for N, but remember, it has a negative sign in front of it in the original N. So, when we use it later, it will be -(16x^2 + 20x + 6), which means -16x^2 - 20x - 6. See why being careful with signs is super important? This kind of meticulousness prevents those tiny errors that can snowball into a completely wrong answer. It’s not just about getting the right answer, but understanding how to get it consistently correct. Always double-check your multiplications, especially when dealing with variables and constants. Mastering this basic expansion is a cornerstone of algebraic proficiency, and it will serve you well in more advanced mathematical concepts.

Tackling the Second Product: (2x+2)(2x-4)

Next up, we have our second product to expand: (2x+2)(2x-4). We'll apply the same FOIL method here to keep things consistent and error-free. Let’s break it down step-by-step:

  • First: Multiply the first terms: (2x) * (2x) = 4x^2.
  • Outer: Multiply the outer terms: (2x) * (-4) = -8x.
  • Inner: Multiply the inner terms: (2) * (2x) = 4x.
  • Last: Multiply the last terms: (2) * (-4) = -8.

Combining these results, we get: 4x^2 - 8x + 4x - 8. Now, let's simplify the x terms: 4x^2 - 4x - 8. Just like before, this expanded form needs to be substituted back into our main expression for N. Again, notice that in the original expression for N, this entire product is preceded by a negative sign: -(2x+2)(2x-4). So, when we substitute, it becomes -(4x^2 - 4x - 8). This means we'll distribute that negative sign across every term inside the parenthesis, resulting in -4x^2 + 4x + 8. This small detail is where many students trip up, so always pause and confirm your signs. It's truly a critical step in avoiding common algebraic mistakes. Practicing these expansions repeatedly builds speed and accuracy, turning what might seem like a daunting task into a routine procedure. Keep up the great work, you're doing awesome!

Putting It All Together: The Grand Simplification

Alright, guys, this is where all our hard work comes together! We have the expanded forms of our products, and now it's time to substitute them back into the original expression for N and simplify. Let’s recall the original N:

N = -(4x+3) - (4x+3)(4x+2) + (2x+1) - (2x+2)(2x-4)

And our expanded products with their respective negative signs already distributed:

  • -(4x+3)(4x+2) became -16x^2 - 20x - 6
  • -(2x+2)(2x-4) became -4x^2 + 4x + 8

Now, let's plug these into N, and include the other single terms: -(4x+3) which simplifies to -4x - 3, and +(2x+1) which is simply 2x + 1.

So, N becomes:

N = (-4x - 3) + (-16x^2 - 20x - 6) + (2x + 1) + (-4x^2 + 4x + 8)

Now, the mission is to combine like terms. Let's group them by power of x:

1. Combine x^2 terms: (-16x^2) + (-4x^2) = -20x^2

2. Combine x terms: (-4x) + (-20x) + (2x) + (4x) = -24x + 6x = -18x

3. Combine constant terms: (-3) + (-6) + (1) + (8) = -9 + 9 = 0

So, after combining everything, our expression for N simplifies to: N = -20x^2 - 18x. Wait, something's not right! The target was N = -6(2x+2). This means I made a mistake in re-transcribing the problem or my initial simplification. Let me re-evaluate the target N-6(2x+2). The original problem statement N-(4x+3) suggests N is defined as (4x+3)-(4x+3)(4x+2)+(2x+1)-(2x+2)(2x-4). If N is the given expression, then the first term N-(4x+3) is not part of the definition of N but rather the first part of the question. Given the structure