Equivalent Expression For 16x^2 + 4: A Math Guide

by TextBrain Team 50 views

Hey guys! Today, we're diving into a fun little math problem where we need to find which expression is equivalent to the polynomial 16x^2 + 4. This type of question often pops up in algebra, and it's super useful to know how to tackle it. So, let's break it down step by step and make sure we understand exactly what's going on.

To start, let's take a closer look at the polynomial itself: 16x^2 + 4. Notice anything special about it? Both terms, 16x^2 and 4, are perfect squares. That's a big clue! When we see perfect squares in a polynomial, it often means we can factor it using some special patterns. One of the most common patterns we use is the difference of squares, but in this case, we have a sum, not a difference. This means we might need to use complex numbers to factor it correctly. Don't worry; it's not as scary as it sounds! We will walk through each step to ensure that we get to the right answer and more importantly, that we understand the underlying principles.

Now, let's consider the given options:

  • A. (4x + 2i)(4x - 2i)
  • B. (4x + 2)(4x - 2)
  • C. (4x + 2)^2
  • D. (4x - 2i)^2

Our mission, should we choose to accept it (and we do!), is to figure out which of these expressions, when multiplied out, gives us back our original polynomial, 16x^2 + 4. We can do this by expanding each option and comparing the result with our target polynomial. This method ensures we’re not just guessing but are actually verifying which expression is the correct equivalent. So, let’s get started and see which one matches!

Step-by-Step Expansion of Options

Option A: (4x + 2i)(4x - 2i)

Let's expand option A, (4x + 2i)(4x - 2i). To do this, we'll use the FOIL method (First, Outer, Inner, Last), which is a handy way to multiply two binomials. This ensures we account for each term in both binomials. So, let's get to it!

  • First: Multiply the first terms in each binomial: (4x)(4x) = 16x^2
  • Outer: Multiply the outer terms: (4x)(-2i) = -8xi
  • Inner: Multiply the inner terms: (2i)(4x) = 8xi
  • Last: Multiply the last terms: (2i)(-2i) = -4i^2

Now, let’s put it all together: 16x^2 - 8xi + 8xi - 4i^2. Notice anything interesting? The -8xi and +8xi terms cancel each other out, which simplifies our expression. But what about that -4i^2 term? Remember that i is the imaginary unit, and i^2 = -1. So, -4i^2 becomes -4(-1) = 4. This is a crucial step because it transforms an imaginary component into a real number, which might just lead us to our answer!

Putting it all together, we have 16x^2 + 4. Bingo! This is exactly the polynomial we started with. So, it looks like option A is our winner. But just to be thorough, let's quickly check the other options to make sure they don't also match. It’s always good to double-check in math, guys, to avoid any sneaky mistakes.

Option B: (4x + 2)(4x - 2)

Okay, let's tackle option B: (4x + 2)(4x - 2). This looks like a classic difference of squares pattern, which is a shortcut we can use. But let's go through the FOIL method again to make sure we see how it works. This time, we’ll be a bit quicker since we’ve already done it once. Using FOIL:

  • First: (4x)(4x) = 16x^2
  • Outer: (4x)(-2) = -8x
  • Inner: (2)(4x) = 8x
  • Last: (2)(-2) = -4

Combining these, we get 16x^2 - 8x + 8x - 4. Just like before, the -8x and +8x terms cancel each other out. This leaves us with 16x^2 - 4. Hmmm, that's close, but not quite what we're looking for. We have a -4 instead of a +4. So, option B is not equivalent to our polynomial.

Option C: (4x + 2)^2

Now, let's examine option C: (4x + 2)^2. This means we're multiplying (4x + 2) by itself: (4x + 2)(4x + 2). Let's use FOIL one more time to expand this binomial. By now, we're practically FOIL experts, right?

  • First: (4x)(4x) = 16x^2
  • Outer: (4x)(2) = 8x
  • Inner: (2)(4x) = 8x
  • Last: (2)(2) = 4

Putting it all together, we have 16x^2 + 8x + 8x + 4, which simplifies to 16x^2 + 16x + 4. Notice that we have an extra 16x term here. This is definitely not the same as our original polynomial, 16x^2 + 4. So, option C is out.

Option D: (4x - 2i)^2

Finally, let's check out option D: (4x - 2i)^2. This means we're multiplying (4x - 2i) by itself: (4x - 2i)(4x - 2i). Let's bring out our FOIL skills one last time and see what we get.

  • First: (4x)(4x) = 16x^2
  • Outer: (4x)(-2i) = -8xi
  • Inner: (-2i)(4x) = -8xi
  • Last: (-2i)(-2i) = 4i^2

Combining these, we get 16x^2 - 8xi - 8xi + 4i^2. Simplifying further, we have 16x^2 - 16xi + 4i^2. Remember that i^2 = -1, so 4i^2 becomes 4(-1) = -4. Our expression now looks like 16x^2 - 16xi - 4. This is not the same as our original polynomial, 16x^2 + 4, because of the -16xi term and the -4 instead of +4. So, option D is also incorrect.

Conclusion: The Correct Expression

After carefully expanding and comparing each option, we found that only one expression is equivalent to the polynomial 16x^2 + 4:

  • A. (4x + 2i)(4x - 2i)

This was a great exercise in using the FOIL method and understanding how complex numbers work in polynomial expressions. By breaking down each option step by step, we were able to confidently identify the correct answer. Remember, in math, it’s all about understanding the process and verifying your results!

So there you have it, folks! We've successfully navigated through this polynomial problem. Keep practicing, and you'll become a math whiz in no time!