**Simplifying rational functions :**

The function which is in the form of f(x) / g(x) is called rational function.

Simplifying rational function is nothing but expressing the the rational function in its lowest term or simplest form.

**Step 1 :**

Factor both numerator and denominator, if possible.

**Step 2 :**

Identify the common factor at both numerator and denominator.

**Step 3 : **

The common factor identified at both numerator and denominator should be multiplied by the other terms.

**Step 4 :**

Now, get rid of the common factor at both numerator and denominator.

The above four steps have been illustrated in the picture given below.

**Example 1 : **

Write the simplified expression for the function defined by

f(x) = 4x**³** / 8x**²**

**Solution: **

In this example we have x**³** and x**²** . We have to split these two terms as much as possible.

f(x) = 4.x.x.x / 8 x.x

Getting rid of two x's at both numerator and denominator, and simplifying 4/8, we we get

f(x) = x / 2

**Example 2 : **

Write the simplified expression for the function defined by

f(x) = (5x + 20) / (7x + 28)

**Solution: **

Factoring the numerator and denominator of the given rational function, we get

f(x) = 5(x + 4) / 7(x + 4)

Getting rid of the common factor (x + 4) at both numerator and denominator, we get

f(x) = 5 / 7

**Example 3 : **

Write the simplified expression for the function defined by

f(x) = (3x + 9) / (3x + 15)

**Solution: **

Factoring the numerator and denominator of the given rational function, we get

f(x) = 3(x + 3) / 3(x + 5)

Getting rid of the common factor 3 at both numerator and denominator, we get

f(x) = (x + 3) / (x + 5)

**Example 4 : **

Write the simplified expression for the function defined by

f(x) = (9x² - 25y²) / (3x² - 5xy)

**Solution: **

f(x) = (9x² - 25y²) / (3x² - 5xy)

f(x) = [(3x)² - (5y)²] / (3x² - 5xy)

Factoring the numerator and denominator of the given rational function, we get

f(x) = [(3x + 5y)(3x - 5y)] / x(3x - 5y)

Getting rid of the common factor (3x - 5y) at both numerator and denominator, we get

f(x) = (3x + 5y) / x

**Example 5 : **

Write the simplified expression for the function defined by

f(x) = (6x² - 54) / (x² + 7x + 12)

**Solution: **

Factoring the numerator and denominator of the given rational function, we get

f(x) = 6(x² - 9) / (x + 3)(x + 4)

f(x) = 6(x² - 3²) / (x + 3)(x + 4)

f(x) = 6(x + 3)(x - 3) / (x + 3)(x + 4)

Getting rid of the common factor (x + 3) at both numerator and denominator, we get

f(x) = 6(x-3) / (x + 4)

**Example 6 : **

Write the simplified expression for the function defined by

f(x) = (64a³ + 125b³) / (4a²b + 5ab²)

**Solution: **

**f(x) = (64a³ + 125b³) / (4a²b + 5ab²) **

**f(x) = (4****³****a³ + 5****³b****³) / (4a²b + 5ab²) **

**f(x) = [(4****a)³ + (5****b)****³] / (4a²b + 5ab²) **

Factoring the numerator and denominator of the given rational function, we get

f(x) = [4a + 5b][(4a)² - 4a.5b + (5b)²] / ab(4a + 5b)

f(x) = [(4a)² - 4a.5b + (5b)²] / ab

f(x) = [16a² - 20ab + 25b²] / ab

Getting rid of the common factor (4a + 5b) at both numerator and denominator, we get

f(x) = [(4a)² - 4a.5b + (5b)²] / ab

f(x) = [16a² - 20ab + 25b²] / ab

**Example 7 : **

Write the simplified expression for the function defined by

f(x) = (x² + 7x + 10) / (x² - 4)

**Solution: **

f(x) = (x² + 7x + 10) / (x² - 4)

f(x) = (x² + 7x + 10) / (x² - 2²)** **

Factoring the numerator and denominator of the given rational function, we get

f(x) = (x + 2)(x + 5) / (x + 2)(x - 2)

Getting rid of the common factor (x + 5) at both numerator and denominator, we get

f(x) = (x + 5) / (x - 2)

After having gone through the stuff given above, we hope that the students would have understood "Simplifying rational functions".

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