Why You Can't Multiply By The LCD And Cancel Terms In Algebraic Fractions

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When dealing with algebraic fractions, it's a common question: Why can't I simply multiply by the Least Common Denominator (LCD) and cancel like-terms as I would with numerical fractions? This query often arises when students encounter problems like the one you've presented. Let's delve into this concept using your example and understand the underlying principles.

Understanding the Problem

You've presented the following problem:

x+4x2+12x+20\frac{x+4}{x^2+12x+20}+x+1x2+8x−20\frac{x+1}{x^2+8x-20}

And you've correctly factored the denominators:

x+4(x+2)(x+10)\frac{x+4}{(x+2)(x+10)} + x+1(x−2)(x+10)\frac{x+1}{(x-2)(x+10)}

The question is, why can't we just multiply by the LCD and cancel terms directly? To answer this, we need to first identify the LCD and then examine the correct procedure for adding algebraic fractions.

Identifying the Least Common Denominator (LCD)

The LCD is the smallest expression that is divisible by both denominators. In this case, the denominators are (x+2)(x+10)(x+2)(x+10) and (x−2)(x+10)(x-2)(x+10). The LCD, therefore, must include all unique factors, which are (x+2)(x+2), (x+10)(x+10), and (x−2)(x-2). Thus, the LCD is:

(x+2)(x+10)(x−2)(x+2)(x+10)(x-2)

Now, let’s explore why directly multiplying by the LCD and canceling terms is not the correct approach.

The Incorrect Approach: Multiplying by the LCD and Cancelling Directly

The temptation to multiply the entire expression by the LCD and cancel terms stems from a misunderstanding of fraction addition. When we add fractions, we need a common denominator so that we can combine the numerators correctly. Multiplying by the LCD across the entire expression changes the value of the expression, which is why it's incorrect.

For instance, if we were to multiply the entire expression by the LCD, we'd be performing an invalid operation. It's akin to changing the equation without maintaining equality. Consider a simpler numerical example:

12+13\frac{1}{2} + \frac{1}{3}

The LCD is 6. If we incorrectly multiply the entire expression by 6, we get:

6∗(12+13)=6∗12+6∗13=3+2=56 * (\frac{1}{2} + \frac{1}{3}) = 6 * \frac{1}{2} + 6 * \frac{1}{3} = 3 + 2 = 5

This is not the same as adding the fractions correctly:

12+13=36+26=56\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}

As you can see, multiplying the entire expression by the LCD changes the result. This same principle applies to algebraic fractions.

The Correct Approach: Finding Equivalent Fractions

The correct way to add fractions, whether numerical or algebraic, is to first find equivalent fractions with the LCD as the denominator. This involves multiplying each fraction by a form of 1 that will result in the LCD in the denominator.

Step-by-Step Solution for the Given Problem

Let's revisit your problem and solve it correctly.

x+4(x+2)(x+10)\frac{x+4}{(x+2)(x+10)} + x+1(x−2)(x+10)\frac{x+1}{(x-2)(x+10)}

The LCD is (x+2)(x+10)(x−2)(x+2)(x+10)(x-2).

  1. Multiply the First Fraction: To get the LCD in the denominator of the first fraction, we need to multiply both the numerator and the denominator by (x−2)(x-2):

    x+4(x+2)(x+10)∗x−2x−2=(x+4)(x−2)(x+2)(x+10)(x−2)\frac{x+4}{(x+2)(x+10)} * \frac{x-2}{x-2} = \frac{(x+4)(x-2)}{(x+2)(x+10)(x-2)}

  2. Multiply the Second Fraction: To get the LCD in the denominator of the second fraction, we need to multiply both the numerator and the denominator by (x+2)(x+2):

    x+1(x−2)(x+10)∗x+2x+2=(x+1)(x+2)(x−2)(x+10)(x+2)\frac{x+1}{(x-2)(x+10)} * \frac{x+2}{x+2} = \frac{(x+1)(x+2)}{(x-2)(x+10)(x+2)}

  3. Combine the Fractions: Now that both fractions have the same denominator, we can add the numerators:

    (x+4)(x−2)(x+2)(x+10)(x−2)+(x+1)(x+2)(x−2)(x+10)(x+2)=(x+4)(x−2)+(x+1)(x+2)(x+2)(x+10)(x−2)\frac{(x+4)(x-2)}{(x+2)(x+10)(x-2)} + \frac{(x+1)(x+2)}{(x-2)(x+10)(x+2)} = \frac{(x+4)(x-2) + (x+1)(x+2)}{(x+2)(x+10)(x-2)}

  4. Expand and Simplify the Numerator: Expand the products in the numerator:

    (x2+2x−8)+(x2+3x+2)(x+2)(x+10)(x−2)\frac{(x^2+2x-8) + (x^2+3x+2)}{(x+2)(x+10)(x-2)}

    Combine like terms:

    2x2+5x−6(x+2)(x+10)(x−2)\frac{2x^2+5x-6}{(x+2)(x+10)(x-2)}

  5. Check for Further Simplification: See if the numerator can be factored to simplify further. In this case, 2x2+5x−62x^2+5x-6 does not factor nicely, so we leave the expression as is.

Thus, the simplified expression is:

2x2+5x−6(x+2)(x+10)(x−2)\frac{2x^2+5x-6}{(x+2)(x+10)(x-2)}

Why This Method Works

The key here is that we are not changing the value of the individual fractions. We are only changing their form. By multiplying both the numerator and denominator by the same expression, we are effectively multiplying by 1, which preserves the fraction's value. This allows us to combine the fractions correctly and simplify the result.

Addressing Mathway's Suggestion

You mentioned that Mathway suggests multiplying the first fraction by (x−2)(x-2). This is precisely the step needed to obtain the LCD in the denominator of the first fraction, as we discussed above. Mathway is guiding you through the correct process of finding equivalent fractions.

Common Mistakes to Avoid

  1. Multiplying the Entire Expression by the LCD: As demonstrated, this changes the value of the expression.
  2. Incorrectly Identifying the LCD: Make sure to include all unique factors from the denominators.
  3. Forgetting to Multiply the Numerator: If you multiply the denominator by a factor, you must also multiply the numerator by the same factor to maintain the fraction's value.
  4. Skipping Simplification: Always simplify the final expression as much as possible.

Best Practices for Algebraic Fractions

  1. Factor Completely: Always factor the denominators (and numerators, if possible) to identify common factors and the LCD.
  2. Identify the LCD: Find the least common denominator by including all unique factors.
  3. Create Equivalent Fractions: Multiply each fraction by the necessary factors to obtain the LCD in the denominator.
  4. Combine Numerators: Add or subtract the numerators once the denominators are the same.
  5. Simplify: Simplify the resulting fraction by canceling common factors.

Conclusion

In conclusion, the reason you can't simply multiply by the LCD and cancel like-terms when adding algebraic fractions is that this changes the value of the expression. The correct method involves finding equivalent fractions with the LCD as the denominator, combining the numerators, and then simplifying. By following this procedure, you ensure that you are performing valid algebraic operations and arriving at the correct solution. Remember, the key is to maintain the value of the expression while manipulating its form to facilitate addition or subtraction. Understanding this principle will help you tackle a wide range of algebraic fraction problems with confidence.

By adhering to these best practices and understanding the underlying principles, you'll be well-equipped to handle algebraic fraction problems accurately and efficiently. The next time you encounter a similar problem, remember to factor, identify the LCD, create equivalent fractions, combine, and simplify. This systematic approach will lead you to the correct solution and deepen your understanding of algebraic manipulations.