Finding Equivalent Expressions: A Math Problem Explained

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Hey math enthusiasts! Today, we're diving into a fun problem involving polynomial functions. We'll break down how to find an equivalent expression when subtracting two functions. It's not as scary as it sounds, I promise! We'll start by defining our functions, and then we'll find out the equivalent expression. So, let's get started. By the end of this, you'll be a pro at this type of problem. Ready? Let's go!

Understanding the Problem: The Basics

First things first, let's get acquainted with the problem. We're given two functions: p(x) and q(x). We know that p(x) = x² - 1 and q(x) = 5(x - 1). The question asks us to figure out which expression is equivalent to (p - q)(x). Now, what does (p - q)(x) even mean? Basically, it means we need to subtract the function q(x) from the function p(x). The result of this subtraction is a new function which is the same as (p - q)(x). Therefore, to solve the problem, we need to subtract the expression that defines q(x) from the expression that defines p(x). We have a set of options and we must choose the correct one. Simple right? Now that we know what to do, let's explore the options and simplify the equation.

Now, let's talk about the options provided. The options present different ways of writing the subtraction. Some of them look right at first glance, but let's carefully check each one to see if they're correct. Remember, the key is to correctly subtract q(x) from p(x). We will substitute the values of the functions p(x) and q(x) into (p - q)(x), and then we will analyze our options. In this process, we must take special care regarding the sign. Because the sign can totally change the result and make the difference.

The Correct Approach to (p−q)(x)(p - q)(x):

To find the equivalent expression for (p - q)(x), we need to perform the subtraction. We know that p(x) = x² - 1 and q(x) = 5(x - 1). So, (p - q)(x) means we have to subtract q(x) from p(x). That would look like this: (x² - 1) - 5(x - 1). See? We're just directly substituting the expressions for the functions and placing a subtraction sign between them. Remember the order is very important, p(x) minus q(x). This is really the heart of the matter. Once we've set up the expression correctly, simplifying it is the next step. Let's see how this aligns with the options we have.

Analyzing the Answer Choices: Step-by-Step

Now, let's take a look at the options one by one, and see which one matches the correct approach. Remember, the correct answer should be equivalent to (x² - 1) - 5(x - 1). Let's see how each option holds up under scrutiny. I suggest you take your time in this process, and not be in a hurry. You've got this!

Option A: (x2−1)−5(x−1)\left(x^2-1\right)-5(x-1)

This option is precisely what we derived when we set up the subtraction: (x² - 1) - 5(x - 1). It correctly represents the subtraction of q(x) from p(x). So, this looks like the correct answer. We are subtracting the entire expression of q(x), and the parenthesis is important. This ensures we are subtracting the complete term.

Option B: (x2−1)−5x−1\left(x^2-1\right)-5 x-1

This option appears similar to the correct expression, but it's missing crucial parentheses. This option presents an important error. It incorrectly subtracts 5x and then subtracts 1, but it doesn't treat 5(x - 1) as a single unit. It fails to correctly distribute the negative sign to both terms within the parenthesis. Because of this, this is not the right choice. Therefore, we can rule out this one pretty fast.

Option C: (5x−1)−(x2−1)(5 x-1)-\left(x^2-1\right)

This option reverses the order of subtraction, subtracting p(x) from 5x - 1. This is not what we are looking for because it asks us to compute (p - q)(x). It should be the expression of q(x) subtracted from the expression of p(x). Thus, this option isn't the correct choice.

Option D: 5(x−1)−x2−15(x-1)-x^2-1

Similar to option B, this one has issues with the signs and the order of operations. It does not subtract x² - 1 from 5(x-1). Moreover, it incorrectly subtracts the two terms instead of subtracting the entire function. So, this option is also wrong.

The Final Verdict: The Correct Answer

After carefully evaluating each option, it's clear that Option A: (x2−1)−5(x−1)\left(x^2-1\right)-5(x-1) is the correct one. This is because it accurately represents the subtraction of q(x) from p(x). Remember, always double-check the order of operations and the correct use of parentheses when dealing with these types of problems. Now that you've followed along, you've conquered a math problem! Keep practicing, and you'll become a pro at these problems in no time. Congratulations! You've successfully navigated through the problem and picked the right answer. Awesome!

Further Practice and Resources

If you found this problem interesting, here are some suggestions to help you practice and improve your skills. You can also explore more resources to solidify your understanding. Practicing similar problems can help you reinforce what you've learned. Here are some ideas:

  • Try Different Functions: Experiment with different polynomial functions for p(x) and q(x). This will help you get used to the concept. Try changing the degree of the polynomials. This will also help you to enhance your ability to do this kind of exercise.
  • Change the Operation: Instead of subtraction, practice addition, multiplication, or division of functions. This will help you see the similarities and differences.
  • Online Quizzes: There are numerous online platforms like Khan Academy or other educational sites that offer quizzes and exercises. These are great to test your knowledge.

Remember, the more you practice, the more comfortable you'll become with these types of problems. Good luck, and keep learning!