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Definitions and Simplifying
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# Dividing Rational Expressions

Dividing rational expressions follows the same procedure as dividing fractions: flip the second fraction over and change the division to multiplication. Then you finish it just like a multiplication problem.

# Example 1

Simplify $\frac{x^2 - 25}{x - 5}\div\frac{x^2 + 5x - 6}{x^2 + 11x + 30}$.

The first step is to "flip" the second fraction and change the division to multiplication.

$$\frac{x^2 - 25}{x - 5}\cdot\frac{x^2 + 11x + 30}{x^2 + 5x - 6}$$

Now we can factor all the polyomials and look for factors that the numerator and denominator have in common. In our expression, that's going to be the two x + 5's, the x - 5's and the x + 6's

$$\frac{(x-5)(x+5)}{x-5}\cdot\frac{(x+6)(x-1)}{(x+5)(x+6)}$$

Canceling the common factors leaves us with:

$$\frac{1}{1}\cdot\frac{x - 1}{1}=x - 1$$

Where x ≠ -6, -5, 5.

# Example 2

Simplify $$\frac{\frac{6x^2 + 7x - 3}{x + 11x + 30}}{\frac{2x^2 - 5x - 12}{x + x - 2}}$$

That looks a little complicated but try thinking of it as "one fraction" divided by "another fraction". In fact, I'll start by rewriting it that way so that it looks more like Example 1.

$$\frac{6x^2 + 7x - 3}{x + 11x + 30}\div\frac{2x^2 - 5x - 12}{x + x - 2}$$

Now we can proceed with flipping the second fraction and changing to division

$$\frac{6x^2 + 7x - 3}{x + 11x + 30}\cdot\frac{x + x - 2}{2x^2 - 5x - 12}$$

Then we can factor the polynomials and pick out the common factors.

$$\frac{(2x+3)(3x - 1)}{(x + 6)(x + 5)}\cdot\frac{(x + 2)(x - 1)}{(2x + 3)(x - 4)}$$

The only common factor is the 2x + 3's so that's all we can elminate.

$$\frac{(3x - 1)(x + 2)(x - 1)}{(x + 6)(x + 5)(x - 4)}=\frac{3x^3 + 2x^2 + 5x + 2}{x^3 + 7x^2 - 14x - 120}$$

Where x ≠ -3/2.

Dyanmic Tutorial - Dividing Rational Expressions

Directions: This solution has 9 steps. To see a description of each step click on the boxes on the left side below. To see the calculations, click on the corresponding box on the right side. Try working out the solution yourself and use the descriptions if you need a hint and the calculations to check your solution.