Simplifying Algebraic Fractions: A Detailed Solution

by Alex Johnson 53 views

In this article, we will break down the process of simplifying a complex algebraic fraction. We will walk through each step, providing clear explanations and helpful tips along the way. This topic often appears in mathematics courses, and mastering it is crucial for success in algebra and beyond. Let's dive into the problem and find the solution!

Understanding the Problem

The initial expression we need to simplify is: 2- rac{x^2+10xy+7y^2}{x^2+4xy+4y^2}. Our goal is to combine the terms and reduce the fraction to its simplest form. This involves finding a common denominator, combining like terms, and potentially factoring to see if any further simplification is possible. Let's look at the step-by-step solution.

Step 1: Finding a Common Denominator

The first step in simplifying this expression is to find a common denominator. We can rewrite the whole number '2' as a fraction with the same denominator as the second term. The denominator of the second term is x2+4xy+4y2x^2 + 4xy + 4y^2. So, we need to rewrite '2' with this denominator. This is a crucial step in combining fractions.

To do this, we multiply 2 by x2+4xy+4y2x2+4xy+4y2\frac{x^2+4xy+4y^2}{x^2+4xy+4y^2}. This gives us:

2βˆ—x2+4xy+4y2x2+4xy+4y2=2(x2+4xy+4y2)x2+4xy+4y22 * \frac{x^2+4xy+4y^2}{x^2+4xy+4y^2} = \frac{2(x^2+4xy+4y^2)}{x^2+4xy+4y^2}

Now, we have a common denominator and can rewrite the original expression as:

2(x2+4xy+4y2)x2+4xy+4y2βˆ’x2+10xy+7y2x2+4xy+4y2\frac{2(x^2+4xy+4y^2)}{x^2+4xy+4y^2} - \frac{x^2+10xy+7y^2}{x^2+4xy+4y^2}

Step 2: Combining the Numerators

Now that we have a common denominator, we can combine the numerators. This involves distributing the '2' in the first term and then subtracting the second numerator. Be careful with the signs when subtracting, as this is a common area for errors. Accuracy in this step is key to getting the correct answer.

First, distribute the 2:

2x2+8xy+8y2x2+4xy+4y2βˆ’x2+10xy+7y2x2+4xy+4y2\frac{2x^2+8xy+8y^2}{x^2+4xy+4y^2} - \frac{x^2+10xy+7y^2}{x^2+4xy+4y^2}

Next, subtract the numerators:

(2x2+8xy+8y2)βˆ’(x2+10xy+7y2)x2+4xy+4y2\frac{(2x^2+8xy+8y^2) - (x^2+10xy+7y^2)}{x^2+4xy+4y^2}

Now, simplify by combining like terms:

2x2+8xy+8y2βˆ’x2βˆ’10xyβˆ’7y2x2+4xy+4y2=x2βˆ’2xy+y2x2+4xy+4y2\frac{2x^2+8xy+8y^2 - x^2 - 10xy - 7y^2}{x^2+4xy+4y^2} = \frac{x^2 - 2xy + y^2}{x^2+4xy+4y^2}

Step 3: Factoring (If Possible)

After combining the terms, the next step is to see if we can factor the numerator and the denominator. Factoring can help us simplify the expression further by canceling out common factors. Factoring is a powerful tool in simplifying algebraic expressions.

Let's factor the numerator, x2βˆ’2xy+y2x^2 - 2xy + y^2. This is a perfect square trinomial, which factors as:

x2βˆ’2xy+y2=(xβˆ’y)2x^2 - 2xy + y^2 = (x - y)^2

Now, let's factor the denominator, x2+4xy+4y2x^2 + 4xy + 4y^2. This is also a perfect square trinomial, which factors as:

x2+4xy+4y2=(x+2y)2x^2 + 4xy + 4y^2 = (x + 2y)^2

So, our expression now looks like:

(xβˆ’y)2(x+2y)2\frac{(x - y)^2}{(x + 2y)^2}

Step 4: Final Simplification

Now that we have factored both the numerator and the denominator, we can see if there are any common factors to cancel out. In this case, there are no common factors between (xβˆ’y)2(x - y)^2 and (x+2y)2(x + 2y)^2. Therefore, the expression is already in its simplest form.

The simplified expression is:

(xβˆ’y)2(x+2y)2\frac{(x - y)^2}{(x + 2y)^2} which is equivalent to x2βˆ’2xy+y2x2+4xy+4y2\frac{x^2-2xy+y^2}{x^2+4xy+4y^2}

Identifying the Correct Answer

Looking back at the given options, we can see that our simplified expression matches option C:

C. x2βˆ’2xy+y2x2+4xy+4y2\frac{x^2-2xy+y^2}{x^2+4xy+4y^2}

Therefore, the correct answer is C.

Common Mistakes to Avoid

When simplifying algebraic fractions, there are several common mistakes that students make. Being aware of these pitfalls can help you avoid them. Prevention is better than cure when it comes to mathematical errors.

  • Incorrectly Distributing the Negative Sign: When subtracting the second fraction, make sure to distribute the negative sign to all terms in the numerator.
  • Forgetting to Find a Common Denominator: You cannot add or subtract fractions without a common denominator.
  • Incorrectly Factoring: Double-check your factoring to ensure it's accurate. A mistake here will lead to an incorrect final answer.
  • Canceling Terms Instead of Factors: You can only cancel out common factors, not individual terms. For example, you cannot cancel out the x2x^2 terms in the original expression before factoring.
  • Not Simplifying Completely: Always check if the final expression can be further simplified. Look for common factors to cancel out.

Tips for Success

To improve your skills in simplifying algebraic fractions, here are some tips:

  • Practice Regularly: The more you practice, the more comfortable you will become with the process.
  • Review Factoring Techniques: A strong understanding of factoring is essential for simplifying fractions.
  • Show Your Work: Write down each step clearly. This makes it easier to spot mistakes.
  • Check Your Answer: If possible, plug in some values for x and y to see if your simplified expression matches the original expression.
  • Understand the Rules: Make sure you understand the rules for adding, subtracting, multiplying, and dividing fractions.

Conclusion

Simplifying algebraic fractions can seem daunting at first, but by breaking it down into manageable steps, it becomes much easier. Remember to find a common denominator, combine the numerators, factor if possible, and simplify completely. By following these steps and avoiding common mistakes, you can master this important mathematical skill. And always remember, practice makes perfect!

For further learning and practice on algebraic fractions, check out resources like Khan Academy. They offer excellent lessons and practice exercises.