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Dividing Rational Expressions

A rational expression is defined as nothing more than a fraction in which the numerator and or the denominator are polynomials. A rational expression is also defined as the division of two polynomials. A dividing rational expression is an algebra expression of the form $\frac{X}{Y}$, where X and Y are simpler expressions. The condition here is that Y should not be zero.

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How to Divide Rational Expressions?

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Given below are the steps for dividing rational expressions:

Step 1: Factorize the rational expression, if possible.

Step 2: Take reciprocal of the second expression.

Step 3: Follow the procedure of multiplication of rational expressions.

Step 4: Reduce it in lowest form. A rational expression has been reduced to it's lowest term only when all the common factors from the numerator and the denominator has been cancelled.

Let us solve $\frac{12x^4}{x - 2}$ $\div $ $\frac{8x^2}{x - 2}$

Given expression is $\frac{12x^4}{x - 2}$ $\div $ $\frac{8x^2}{x - 2}$

Step 1:
Change $\div $ sign into * sign.

$\frac{12x^4}{x - 2}$ * $\frac{x - 2}{8x^2}$

Step 2:
Multiply numerators and denominators separately:

$\frac{(12x^4)(x - 2)}{8x^2(x - 2)}$


Step 3:
Reduce it in the simplest form. Common factors from the numerator and the denominator can be cancelled.

$\frac{(3x^2)}{2}$

Dividing Rational Expressions Examples

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Below you can see how to divide Rational Expressions with some examples.

Solved Examples

Question 1: Simplify the following expressions

$\frac{56(x + 28)}{y}$ $\div$ $\frac{x + 28}{y}$
Solution:
Given $\frac{56(x + 28)}{y}$ $\div$ $\frac{x + 28}{y}$

Step 1:
Reciprocal the values and cross multiply

= $\frac{56(x + 28)}{y}$ $\times$ $\frac{y}{x + 28}$


Step 2: Cancelling common terms, we have

= $\frac{56(x + 28)}{y}$ $\times$ $\frac{y}{x + 28}$

= 56

Question 2: Solve the following expression:
$\frac{x^2 + 2x - 15}{x^2 - 4x - 45}$$\div$ $\frac{x^2 + x - 12}{x^2 - 5x - 36}$
Solution:
We need to factor the numerators and the denominators of both the rational expressions.

The factors of the first rational expression are

Numerator = x2 + 2x - 15 = (x + 5)(x - 3)

Denominator = x2 - 4x - 45 = (x - 9)(x + 5)

The factors of the second rational expression are

Numerator = x2 + x - 12 = (x + 4)(x - 3)

Denominator = x2 - 5x - 36 = (x - 9) (x + 4)

Now $\frac{x^2 + 2x - 15}{x^2 - 4x - 45}$$\div$ $\frac{x^2 + x - 12}{x^2 - 5x - 36}$ has been reduced to $\frac{(x+5)(x-3)}{(x-9)(x+5)}$ $\div$ $\frac{(x+4)(x-3)}{ (x-9)(x+4)}$

= $\frac{(x+5)(x-3)}{(x-9)(x+5)}$ * $\frac{(x-9)(x+4)}{(x+4)(x-3)}$

Cancel the common terms

= 1



Question 3: Simplify the following expressions

$\frac{11p}{p + 4}$ $\div$ $\frac{p + 4}{p}$
Solution:

Given $\frac{11p}{p + 4}$ $\div$ $\frac{p + 4}{p}$

Step 1: Reciprocal the values and cross multiply

= $\frac{11p}{p + 4}$ $\times$ $\frac{p}{p + 4}$

Step 2: Cancelling common terms, we have

= $\frac{11p}{p + 4}$ $\times$ $\frac{p}{p + 4}$

= $\frac{11 p^2}{(p + 4)^2}$



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