The Easy Way to Add Algebraic Fractions 2

Addition of Algebraic Fractions Monic Expressions

Addition of Algebraic Fractions: Difference of Squares

As an experienced mathematics tutor, I have noticed that many students struggle with adding algebraic fractions, particularly those involving the difference of squares. In this article, I will provide a step-by-step guide to simplify the process and help you master this essential skill.

Understanding the Difference of Squares

The difference of squares is a special case of polynomial factorisation. It states that the difference of two perfect squares can be factored as the product of their sum and difference. The general form is:

\( \displaystyle a^2 – b^2 = (a+b)(a-b) \)

This concept is crucial when adding algebraic fractions with denominators that are the difference of squares.

Example: Adding Algebraic Fractions with the Difference of Squares

Let’s consider the following example:

\( \displaystyle \frac{2}{x^2-16} + \frac{3}{16-4x} \)

To add these fractions, we need to find a common denominator. In this case, both denominators are the difference of squares, which we can factorise to simplify the process.

Step 1: Factorise the Denominators

Factorise the first denominator, $x^2-16$, using the difference of squares formula:

\( \displaystyle x^2-16 = (x+4)(x-4) \)

Now, factorise the second denominator, $16-4x$ by its common factor.

\( \displaystyle 16-4x = 4(4-x) = -4(x-4) \)

Step 2: Find the Least Common Denominator (LCD)

The LCD is the smallest polynomial that is divisible by all the denominators of the fractions being added. In this case, the LCD is:

\( \displaystyle 4(x+4)(x-4) \)

Step 3: Convert Each Fraction to an Equivalent Fraction with the LCD

To add the fractions, they must have the same denominator. Convert each fraction to an equivalent fraction with the LCD as the denominator.

For the first fraction:

\( \begin{align} \displaystyle \frac{2}{x^2-16} &= \frac{2}{(x+4)(x-4)} \\ &= \frac{2}{(x+4)(x-4)} \times \frac{4}{4} \\ &= \frac{8}{4(x+4)(x-4)} \end{align} \)

For the second fraction:

\( \begin{align} \displaystyle \frac{3}{16-4x} &= \frac{3}{4(4-x)} \\ &= \frac{-3}{4(x-4)} \\ &= \frac{-3}{4(x-4)} \times \frac{x+4}{x+4} \\ &= \frac{-3(x+4)}{4(x-4)(x+4)} \end{align} \)

Step 4: Add the Numerators and Simplify

Now that the fractions have the same denominator, add the numerators and keep the common denominator:

\( \begin{align} \displaystyle \frac{2}{x^2-16} + \frac{3}{16-4x} &= \frac{8}{4(x+4)(x-4)} + \frac{-3(x+4)}{4(x-4)(x+4)} \\ &= \frac{8-3(x+4)}{4(x-4)(x+4)} \\ &= \frac{-3x-4}{4(x-4)(x+4)} \end{align} \)

Key Points to Remember

  1. The difference of squares is a special case of polynomial factorisation, where $a^2-b^2 = (a+b)(a-b)$.
  2. When adding algebraic fractions with denominators that are the difference of squares, factorise the denominators first.
  3. Find the least common denominator (LCD) by considering all the factors of the denominators.
  4. Convert each fraction to an equivalent fraction with the LCD as the denominator.
  5. Add the numerators and simplify the result.

Practice Makes Perfect

Adding algebraic fractions with the difference of squares may seem challenging at first, but with practice, it becomes easier. Remember to follow the key steps outlined in this article and take your time to avoid mistakes.

Additional Tips

  • Double-check your factorisation to ensure accuracy.
  • Simplify the numerator and denominator as much as possible to obtain the most simplified result.
  • If you’re stuck, try breaking down the problem into smaller, manageable steps.

By mastering the art of adding algebraic fractions involving the difference of squares, you’ll be well-prepared to tackle more advanced mathematical concepts. Remember, patience and perseverance are essential for success in mathematics.

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