Multiplying And Simplifying Rational Expressions A Step-by-Step Guide

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In this article, we will delve into the process of multiplying rational expressions and simplifying the result. We will tackle the following expression:

3x2+2x−21−2x2−2x+12⋅2x2+25x+636x2+7x−49\frac{3 x^2+2 x-21}{-2 x^2-2 x+12} \cdot \frac{2 x^2+25 x+63}{6 x^2+7 x-49}

Furthermore, we'll explore how to determine the values of constants b, c, and d when a = 1, such that the simplified expression is equivalent to a given form. This involves factoring quadratic expressions, identifying common factors for cancellation, and understanding the conditions under which the expressions are undefined. Let's embark on this mathematical journey step by step.

Step-by-Step Multiplication and Simplification

To effectively multiply these expressions, we will break down each step in detail. This approach ensures a clear understanding of the process, from factoring to simplifying.

1. Factor Each Quadratic Expression

The first crucial step in simplifying rational expressions is to factor each quadratic expression present. Factoring allows us to identify common factors between the numerator and the denominator, which can then be cancelled out. This simplification is key to arriving at the most reduced form of the expression. Let's factor each quadratic expression individually:

  • 3x² + 2x - 21: To factor this quadratic, we need to find two numbers that multiply to (3)(-21) = -63 and add up to 2. These numbers are 9 and -7. So, we can rewrite the middle term as 9x - 7x:

    3x² + 9x - 7x - 21 = 3x(x + 3) - 7(x + 3) = (3x - 7)(x + 3)

  • -2x² - 2x + 12: First, factor out a -2: -2(x² + x - 6). Now, we need two numbers that multiply to -6 and add up to 1. These numbers are 3 and -2. So, the factored form is:

    -2(x² + 3x - 2x - 6) = -2[x(x + 3) - 2(x + 3)] = -2(x - 2)(x + 3)

  • 2x² + 25x + 63: We look for two numbers that multiply to (2)(63) = 126 and add up to 25. These numbers are 18 and 7. Rewriting the middle term:

    2x² + 18x + 7x + 63 = 2x(x + 9) + 7(x + 9) = (2x + 7)(x + 9)

  • 6x² + 7x - 49: Find two numbers that multiply to (6)(-49) = -294 and add up to 7. These numbers are 21 and -14. Rewriting the middle term:

    6x² + 21x - 14x - 49 = 3x(2x + 7) - 7(2x + 7) = (3x - 7)(2x + 7)

2. Rewrite the Expression with Factored Forms

Now that we have factored each quadratic expression, we can rewrite the original expression using these factored forms. This step is crucial because it sets the stage for the cancellation of common factors, which will ultimately simplify the expression. Substituting the factored forms into the original expression, we get:

(3x−7)(x+3)−2(x−2)(x+3)⋅(2x+7)(x+9)(3x−7)(2x+7)\frac{(3x - 7)(x + 3)}{-2(x - 2)(x + 3)} \cdot \frac{(2x + 7)(x + 9)}{(3x - 7)(2x + 7)}

This rewritten expression clearly displays the factors in both the numerators and denominators, making it easier to identify common terms that can be cancelled. The next step involves carefully examining these factors to perform the cancellations.

3. Cancel Common Factors

With the expression rewritten in its factored form, we can now identify and cancel common factors that appear in both the numerator and the denominator. This process of cancellation is fundamental to simplifying rational expressions, as it reduces the expression to its most basic form. By carefully examining the factored expression:

(3x−7)(x+3)−2(x−2)(x+3)⋅(2x+7)(x+9)(3x−7)(2x+7)\frac{(3x - 7)(x + 3)}{-2(x - 2)(x + 3)} \cdot \frac{(2x + 7)(x + 9)}{(3x - 7)(2x + 7)}

We can observe the following common factors:

  • (3x - 7) appears in both the numerator and the denominator.
  • (x + 3) appears in both the numerator and the denominator.
  • (2x + 7) appears in both the numerator and the denominator.

Cancelling these common factors, we simplify the expression as follows:

(3x−7)(x+3)−2(x−2)(x+3)⋅(2x+7)(x+9)(3x−7)(2x+7)=x+9−2(x−2)\frac{\cancel{(3x - 7)}\cancel{(x + 3)}}{-2(x - 2)\cancel{(x + 3)}} \cdot \frac{\cancel{(2x + 7)}(x + 9)}{\cancel{(3x - 7)}\cancel{(2x + 7)}} = \frac{x + 9}{-2(x - 2)}

This simplified expression is much cleaner and easier to work with than the original. However, it's essential to remember that this simplification is valid only for values of x that do not make the original expression undefined. This leads us to the next critical step: identifying the values of x that must be excluded.

4. Simplify the Expression

After cancelling the common factors, we are left with the simplified form of the expression. This simplified form is much easier to work with and provides a clearer representation of the original expression's behavior. From the previous step, we have:

x+9−2(x−2)\frac{x + 9}{-2(x - 2)}

This can be further simplified by distributing the -2 in the denominator:

x+9−2x+4\frac{x + 9}{-2x + 4}

This is the simplified form of the original expression. However, it's important to note that this simplification is valid only for values of x that do not make the original expression undefined. We need to consider the values of x that would make any of the denominators in the original expression equal to zero. These values must be excluded from the domain of the simplified expression.

5. Identify Values of x that Make the Expression Undefined

To ensure the simplified expression is mathematically sound, we must identify any values of x that would make the original expression undefined. A rational expression is undefined when its denominator is equal to zero. Therefore, we need to find the values of x that make the denominators of the original expression equal to zero. The denominators in the original expression are:

  • -2x² - 2x + 12
  • 6x² + 7x - 49

We already factored these expressions in the first step, so we can use the factored forms to find the values of x that make them zero:

  • -2(x - 2)(x + 3) = 0 => x = 2 or x = -3
  • (3x - 7)(2x + 7) = 0 => x = 7/3 or x = -7/2

Therefore, the original expression is undefined when x is 2, -3, 7/3, or -7/2. These values must be excluded from the domain of the simplified expression. This ensures that the simplified expression is equivalent to the original expression for all valid values of x.

Determine Constants b, c, and d

Now, let's address the second part of the problem: finding the values of the constants b, c, and d when a = 1, such that the simplified expression is equivalent to a given form. While the specific form isn't provided in the problem statement, we can illustrate the process with a general example. Let's assume we want to express the simplified expression in the form:

x+9−2x+4=x+9ax+b=x+cdx+4\frac{x + 9}{-2x + 4} = \frac{x + 9}{ax + b} = \frac{x + c}{dx + 4}

Given that a = 1, we can rewrite the first equation as:

x+9−2x+4=x+9x+b\frac{x + 9}{-2x + 4} = \frac{x + 9}{x + b}

From this, it's clear that there is no value of b that would make the denominators equivalent, as -2x + 4 cannot be transformed into x + b by simply adjusting b. This indicates a potential error in the problem statement or a misunderstanding of the intended form.

However, if we focus on the second equation:

x+9−2x+4=x+cdx+4\frac{x + 9}{-2x + 4} = \frac{x + c}{dx + 4}

We can equate the denominators to find d:

-2x + 4 = dx + 4

This implies that d = -2. Now, to find c, we can cross-multiply:

(x + 9)(dx + 4) = (x + c)(-2x + 4)

Substituting d = -2:

(x + 9)(-2x + 4) = (x + c)(-2x + 4)

Since the denominators are the same, we can equate the numerators:

x + 9 = x + c

This implies that c = 9.

Therefore, in this example, if the target form is x+cdx+4{\frac{x + c}{dx + 4}}, then c = 9 and d = -2.

General Approach for Determining Constants

In general, to determine the values of constants that make two rational expressions equivalent, follow these steps:

  1. Simplify both expressions: Factor and cancel common factors.
  2. Equate the expressions: Set the two expressions equal to each other.
  3. Cross-multiply: Multiply the numerator of the first expression by the denominator of the second expression and vice versa.
  4. Equate coefficients: Compare the coefficients of the corresponding terms on both sides of the equation.
  5. Solve for the constants: Solve the resulting system of equations for the unknown constants.

This approach allows you to systematically determine the values of constants that ensure the equivalence of two rational expressions.

Conclusion

In this article, we've thoroughly explored the process of multiplying rational expressions, simplifying them, and identifying values that make the expression undefined. We've also addressed the method for determining constants that make two rational expressions equivalent. By following these steps, you can confidently tackle similar problems and deepen your understanding of rational expressions.

Key Takeaways:

  • Factoring is essential for simplifying rational expressions.
  • Cancelling common factors reduces the expression to its simplest form.
  • Identifying values that make the expression undefined is crucial for mathematical accuracy.
  • Equating coefficients allows you to solve for unknown constants.
  • Understanding the steps involved ensures a systematic approach to problem-solving.

By mastering these concepts, you'll be well-equipped to handle a wide range of problems involving rational expressions. Remember to practice regularly and apply these techniques to various examples to solidify your understanding.