Simplifying Rational Expressions A Step By Step Guide

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In this article, we will walk through the step-by-step process of simplifying the given algebraic expression. This involves combining fractions with common denominators, simplifying the numerator, factoring quadratic expressions, and canceling common factors. We aim to provide a comprehensive guide suitable for students and anyone looking to enhance their algebra skills. Let’s delve into the details and make this complex expression simpler.

Understanding the Problem

Before diving into the solution, let's clearly state the expression we need to simplify:

(5m^2 - 13m + 2) / (m^2 - 64) - (4m^2 - 7m + 18) / (m^2 - 64)

This expression involves two rational functions with the same denominator. The first step in simplifying such expressions is to combine the numerators over the common denominator. This allows us to consolidate terms and potentially identify opportunities for factoring and simplification. The denominator m^2 - 64 is a difference of squares, which we can factor as (m - 8)(m + 8). Recognizing this structure early on is crucial, as it may lead to cancellations later in the process. Our goal is to reduce the expression to its simplest form, where no further simplification is possible.

To successfully simplify this expression, we need to be comfortable with several key algebraic techniques. These include:

  1. Combining like terms in the numerator.
  2. Factoring quadratic expressions.
  3. Recognizing and factoring the difference of squares.
  4. Canceling common factors between the numerator and the denominator.

Each of these steps requires careful attention to detail and a solid understanding of algebraic principles. Let's begin by combining the numerators and simplifying the resulting expression.

Step 1: Combine the Numerators

The first step in simplifying the expression is to combine the numerators since the fractions share a common denominator. We subtract the second numerator from the first:

(5m^2 - 13m + 2) - (4m^2 - 7m + 18)

This subtraction needs to be performed carefully, distributing the negative sign across the second set of parentheses. This is a crucial step to avoid sign errors, which are common pitfalls in algebraic manipulations. Distributing the negative sign, we get:

5m^2 - 13m + 2 - 4m^2 + 7m - 18

Now, we combine like terms. This involves grouping terms with the same power of m together and adding or subtracting their coefficients. We have 5m^2 and -4m^2 terms, -13m and 7m terms, and constant terms 2 and -18. Combining these, we get:

(5m^2 - 4m^2) + (-13m + 7m) + (2 - 18)

Performing the arithmetic, we simplify to:

m^2 - 6m - 16

So, the simplified numerator is m^2 - 6m - 16. Now we place this over the common denominator, which gives us:

(m^2 - 6m - 16) / (m^2 - 64)

This new expression is a significant step towards simplification. Next, we need to factor both the numerator and the denominator to see if there are any common factors that can be canceled. Factoring the quadratic expression in the numerator is the next key step, and we'll approach this methodically to ensure accuracy.

Step 2: Factor the Numerator and Denominator

Now that we have the expression (m^2 - 6m - 16) / (m^2 - 64), the next crucial step is to factor both the numerator and the denominator. Factoring allows us to identify common factors that can be canceled, further simplifying the expression.

Factoring the Numerator

The numerator is a quadratic expression of the form m^2 - 6m - 16. To factor this, we look for two numbers that multiply to -16 and add up to -6. Let’s list the factor pairs of -16:

  • 1 and -16
  • -1 and 16
  • 2 and -8
  • -2 and 8
  • 4 and -4

Among these pairs, 2 and -8 add up to -6. Thus, we can factor the numerator as:

m^2 - 6m - 16 = (m + 2)(m - 8)

This factorization is a critical step, as it reveals one of the factors that might cancel with the denominator.

Factoring the Denominator

The denominator is m^2 - 64, which is a difference of squares. Recognizing the difference of squares pattern is essential here. The general form for a difference of squares is a^2 - b^2 = (a - b)(a + b). In this case, a = m and b = 8, so we can factor the denominator as:

m^2 - 64 = (m - 8)(m + 8)

Now that we have factored both the numerator and the denominator, our expression looks like this:

((m + 2)(m - 8)) / ((m - 8)(m + 8))

Next, we identify and cancel common factors to further simplify the expression. This step is the heart of simplifying rational expressions, and it requires careful attention to ensure we only cancel factors that are truly common.

Step 3: Cancel Common Factors

Having factored the numerator and the denominator, our expression stands as:

((m + 2)(m - 8)) / ((m - 8)(m + 8))

Now, we look for common factors in both the numerator and the denominator. We can see that the term (m - 8) appears in both. Canceling this common factor is the key to simplifying the expression.

When we cancel the (m - 8) terms, we are left with:

(m + 2) / (m + 8)

This resulting expression is significantly simpler than the original. It’s crucial to understand that we can only cancel factors, not terms. In other words, we can cancel expressions that are multiplied, but not those that are added or subtracted separately. This is a common mistake in algebra, so it’s worth emphasizing.

The simplified expression (m + 2) / (m + 8) has no further common factors between the numerator and the denominator. This means we have successfully simplified the original expression to its simplest form. We cannot cancel the m terms or the constants because they are not factors of the entire numerator or denominator; they are terms within the expressions (m + 2) and (m + 8).

At this stage, it's always a good practice to review our steps and ensure no further simplification is possible. In this case, the expression (m + 2) / (m + 8) is indeed in its simplest form.

Final Answer

After carefully combining numerators, factoring quadratic expressions, and canceling common factors, we have successfully simplified the given expression. The final simplified form is:

(m + 2) / (m + 8)

This concise expression is equivalent to the original complex expression, but much easier to understand and work with. By breaking down the problem into smaller steps and applying algebraic principles methodically, we were able to simplify it effectively.

Summary of Steps

  1. Combine the Numerators: Since the fractions had a common denominator, we subtracted the second numerator from the first and simplified the result to m^2 - 6m - 16.
  2. Factor the Numerator and Denominator: We factored the numerator m^2 - 6m - 16 into (m + 2)(m - 8) and the denominator m^2 - 64 into (m - 8)(m + 8).
  3. Cancel Common Factors: We canceled the common factor (m - 8) from both the numerator and the denominator.
  4. Final Simplified Expression: The simplified expression is (m + 2) / (m + 8).

This process highlights the importance of understanding fundamental algebraic techniques such as factoring, combining like terms, and recognizing patterns like the difference of squares. With these skills, complex algebraic expressions can be simplified into manageable forms.

In conclusion, simplifying algebraic expressions is a skill that builds on a strong foundation of algebraic principles. By mastering these principles and practicing methodical approaches, anyone can tackle complex problems with confidence. The final answer, (m + 2) / (m + 8), represents the simplified form of the original expression, achieved through careful algebraic manipulation.