Simplifying Expressions With Rational Functions P(x) And Q(x)

Jul 12, 2025 by ADMIN 62 views

Exploring Rational Functions P(x) and Q(x) and Their Simplified Expressions

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In the realm of mathematical functions, rational functions hold a significant position. They are defined as the ratio of two polynomials, and their behavior can be quite intriguing. In this article, we will delve into two specific rational functions, P(x) and Q(x), and explore their properties and simplified forms. We will analyze the expressions formed by combining these functions and match them with their corresponding simplified forms. This exploration will enhance our understanding of rational functions and their manipulations.

Defining the Rational Functions P(x) and Q(x)

Let's begin by defining the two rational functions that will be the focus of our discussion. We have:

P(x) = 2 / (3x - 1)
Q(x) = 6 / (-3x + 2)

These functions are rational because they are expressed as the ratio of two polynomials. In this case, the numerator of P(x) is the constant polynomial 2, and the denominator is the linear polynomial 3x - 1. Similarly, the numerator of Q(x) is the constant polynomial 6, and the denominator is the linear polynomial -3x + 2. Understanding these functions individually is the first step towards comprehending their combined behavior.

Analyzing P(x) = 2 / (3x - 1)

To gain a deeper understanding of P(x), let's analyze its key characteristics. The denominator, 3x - 1, plays a crucial role in determining the function's behavior. Specifically, we need to identify the values of x for which the denominator becomes zero, as these values will result in the function being undefined. Setting 3x - 1 = 0, we find that x = 1/3. This means that P(x) has a vertical asymptote at x = 1/3. A vertical asymptote is a vertical line that the graph of the function approaches but never touches.

Furthermore, we can analyze the behavior of P(x) as x approaches positive and negative infinity. As x becomes very large (positive or negative), the term 3x dominates the denominator, and the function approaches 0. This indicates that P(x) has a horizontal asymptote at y = 0. A horizontal asymptote is a horizontal line that the graph of the function approaches as x approaches infinity or negative infinity. Understanding the asymptotes helps us visualize the overall shape and behavior of the function.

Analyzing Q(x) = 6 / (-3x + 2)

Now, let's turn our attention to the function Q(x) = 6 / (-3x + 2). Similar to our analysis of P(x), we need to identify the values of x that make the denominator zero. Setting -3x + 2 = 0, we find that x = 2/3. This means that Q(x) has a vertical asymptote at x = 2/3. The function will approach positive or negative infinity as x gets closer to 2/3.

We can also analyze the horizontal asymptote of Q(x). As x approaches positive or negative infinity, the term -3x in the denominator dominates, and the function approaches 0. Therefore, Q(x) also has a horizontal asymptote at y = 0. Understanding the asymptotes of both P(x) and Q(x) is essential for predicting the behavior of expressions involving these functions.

Matching Expressions with Their Simplified Forms

Now that we have a solid understanding of the individual functions P(x) and Q(x), we can move on to the core task of this article: matching expressions involving these functions with their simplified forms. The expression we are given is:

2(12x + 1) / ((3x - 1)(-3x + 2))

Our goal is to determine how this expression relates to the functions P(x) and Q(x) and whether it can be further simplified. This involves algebraic manipulation and a careful consideration of the properties of rational functions.

Breaking Down the Expression

To begin, let's examine the structure of the given expression. The numerator is 2(12x + 1), which is a linear expression. The denominator is (3x - 1)(-3x + 2), which is the product of the denominators of P(x) and Q(x). This observation suggests that the expression might be related to some combination of P(x) and Q(x). The key is to manipulate the expression algebraically to reveal its connection to the original functions.

Expanding the denominator, we get:

(3x - 1)(-3x + 2) = -9x^2 + 6x + 3x - 2 = -9x^2 + 9x - 2

So, the expression becomes:

2(12x + 1) / (-9x^2 + 9x - 2)

Now, we need to investigate how this expression can be related back to P(x) and Q(x).

Connecting the Expression to P(x) and Q(x)

Recall the definitions of P(x) and Q(x):

P(x) = 2 / (3x - 1)
Q(x) = 6 / (-3x + 2)

Notice that the denominators of P(x) and Q(x) appear as factors in the denominator of our expression. This suggests that we might be able to express the given expression as a sum or difference of P(x) and Q(x), or some multiple thereof. To explore this possibility, let's consider adding P(x) and Q(x):

P(x) + Q(x) = 2 / (3x - 1) + 6 / (-3x + 2)

To add these fractions, we need a common denominator, which is the product of the individual denominators:

Common denominator = (3x - 1)(-3x + 2) = -9x^2 + 9x - 2

Now, we can rewrite the sum with the common denominator:

P(x) + Q(x) = [2(-3x + 2) + 6(3x - 1)] / (-9x^2 + 9x - 2)

Simplifying the Sum P(x) + Q(x)

Let's simplify the numerator of the sum:

2(-3x + 2) + 6(3x - 1) = -6x + 4 + 18x - 6 = 12x - 2

So, we have:

P(x) + Q(x) = (12x - 2) / (-9x^2 + 9x - 2)

Now, let's compare this with the original expression:

Original expression: 2(12x + 1) / (-9x^2 + 9x - 2)
P(x) + Q(x) = (12x - 2) / (-9x^2 + 9x - 2)

We can see that the denominators match, but the numerators are different. The original expression has a numerator of 2(12x + 1) = 24x + 2, while P(x) + Q(x) has a numerator of 12x - 2. This indicates that the original expression is not simply P(x) + Q(x).

Exploring Other Combinations

Since the original expression is not a direct sum of P(x) and Q(x), we might consider other combinations, such as multiples of P(x) and Q(x). Let's try multiplying P(x) + Q(x) by a constant to see if we can match the numerator of the original expression. Suppose we multiply by a constant k:

k[P(x) + Q(x)] = k(12x - 2) / (-9x^2 + 9x - 2)

We want to find a value of k such that:

k(12x - 2) = 2(12x + 1)

12kx - 2k = 24x + 2

Equating the coefficients of x, we get:

12k = 24  =>  k = 2

Equating the constant terms, we get:

-2k = 2  =>  k = -1

Since we have conflicting values for k, it is impossible to express the original expression as a constant multiple of P(x) + Q(x). This suggests that the original expression might be a more complex combination of P(x) and Q(x).

Reassessing the Original Expression

Let's take another look at the original expression:

2(12x + 1) / ((3x - 1)(-3x + 2))

We know that the denominator is the product of the denominators of P(x) and Q(x). The numerator, 2(12x + 1), does not seem to have a direct relationship to the numerators of P(x) and Q(x). However, let's try a different approach. Instead of trying to express the given expression as a sum or multiple of P(x) and Q(x), let's consider if it can be simplified by partial fraction decomposition. While partial fraction decomposition is typically used when we have a sum of rational expressions, it can also help us understand the structure of a single complex rational expression.

Partial Fraction Decomposition (Alternative Approach)

We can rewrite the given expression as:

(24x + 2) / ((3x - 1)(-3x + 2))

We want to express this in the form:

A / (3x - 1) + B / (-3x + 2)

Where A and B are constants. Multiplying both sides by the denominator (3x - 1)(-3x + 2), we get:

24x + 2 = A(-3x + 2) + B(3x - 1)

24x + 2 = -3Ax + 2A + 3Bx - B

Now, we equate the coefficients of x and the constant terms:

-3A + 3B = 24

2A - B = 2

We now have a system of two linear equations with two variables. We can simplify the first equation by dividing by 3:

-A + B = 8

2A - B = 2

Adding the two equations, we get:

A = 10

Substituting A = 10 into the first equation:

-10 + B = 8

B = 18

So, we have A = 10 and B = 18. Therefore, the original expression can be rewritten as:

10 / (3x - 1) + 18 / (-3x + 2)

Final Simplification and Matching

Now, let's express this in terms of P(x) and Q(x). Recall:

P(x) = 2 / (3x - 1)
Q(x) = 6 / (-3x + 2)

We can rewrite the expression as:

10 / (3x - 1) = 5 * [2 / (3x - 1)] = 5P(x)

18 / (-3x + 2) = 3 * [6 / (-3x + 2)] = 3Q(x)

Therefore, the simplified form of the original expression is:

5P(x) + 3Q(x)

This is the simplified form of the expression. By systematically breaking down the problem and exploring different algebraic manipulations, we have successfully matched the given expression with its simplified form.

Conclusion

In this article, we explored the rational functions P(x) = 2 / (3x - 1) and Q(x) = 6 / (-3x + 2). We analyzed their individual properties, including their vertical and horizontal asymptotes. We then tackled the challenge of matching a given expression, 2(12x + 1) / ((3x - 1)(-3x + 2)), with its simplified form. Through a series of algebraic manipulations, including partial fraction decomposition, we determined that the simplified form is 5P(x) + 3Q(x). This exercise demonstrates the importance of understanding the properties of rational functions and the power of algebraic techniques in simplifying complex expressions. Mastering these skills is crucial for success in advanced mathematics and related fields. The ability to analyze and manipulate rational functions opens doors to solving a wide range of problems in various scientific and engineering disciplines.