Mastering the Difference of Squares: A Simple Guide

What is the completely factored form of the expression 4x^2-16?

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

Answer: (C)

To find the completely factored form of the expression 4x^2 - 16, one approach is to first identify that this expression is a difference of squares. The expression can be rewritten as: 4(x^2 - 4) The term x^2 - 4 is itself a difference of squares, which can be factored further as: x^2 - 4 = (x - 2)(x + 2) Now, substituting back into the factored form of the original expression gives us: 4(x - 2)(x + 2) This shows that the completely factored form of the expression 4x^2 - 16 is indeed 4(x - 2)(x + 2). This confirms that the correct answer highlights the process of breaking down the factors correctly and demonstrates the use of the difference of squares technique. The other choices either include incorrect factors or do not represent the correct breakdown of the original expression. For example, while some options might involve factoring out a common factor or using different approaches, they do not lead to the simplest form which is clearly represented by the identified correct answer.

When it comes to preparing for the Postsecondary Education Readiness Test (PERT), you might feel overwhelmed by the various topics you need to tackle. But here’s a handy tip to boost your understanding: mastering the difference of squares can be both enlightening and fun! You know what? Let’s break this down together, starting with the expression 4x² - 16.

Now, if you’re staring at this problem and thinking, “How on earth do I even start?” don’t worry! Recognizing it as a difference of squares is your first big win. Why? Because the difference of squares is like a secret shortcut in algebra that makes life a whole lot easier.

So, what does difference of squares mean, exactly? Well, it refers to an expression in the form of a² - b², which can be factored into (a - b)(a + b). In our case, 4x² - 16 fits the bill perfectly! It’s worth mentioning that 4 can be rewritten as (2)², and 16 is (4)². But let’s stick to what we know: it’s the difference of squares we’re concentrating on here.

Let’s factor it out step by step:

1. Identify the expression: We have 4(x² - 4).

2. Recognize the inner difference of squares: The term x² - 4 can also be factored because it’s (x - 2)(x + 2).

Now, if we pull everything together, we substitute our factored form back into the original setup. What do we get? Ta-da! 4(x - 2)(x + 2). Voilà, we’ve got the completely factored form! Isn’t that just gratifying?

Now, let’s take a moment to pause. Why is it crucial to understand the difference of squares here? Well, it doesn’t just help with this particular expression—grasping this concept can unlock a treasure trove of algebra problems you’ll face in the PERT and beyond!

But you might be wondering why the other options didn't quite cut it. Choices like 4(x² - 4) or 2(x - 4)(2x + 4) just don’t accurately reflect how we break down our expression. They either include incorrect factors or miss the most efficient factored form.

This simple yet powerful technique not only prep you for the PERT but also helps solidify your overall understanding of algebra. I mean, think about it: wouldn’t it be nice to feel confident and in control while tackling your math studies?

After all, every little victory—like grasping the difference of squares—takes you one step closer to acing that test. Remember, practice can’t be stressed enough, so keep working through problems like these. You’ll not only improve your skills but may find some enjoyment in uncovering the patterns hidden within equations.

Before long, you’ll be ready to face the PERT with a sense of calm and assurance that you’ve got this. Happy studying!