**Simplifying rational functions worksheet :**

Worksheet on simplifying rational functions is much useful to the students who would like to practice problems on rational expressions and functions.

Write the simplified expression for the functions given below.

1) f(x) = 4x**³** / 8x**²**

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

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

4) f(x) = (9x² - 25y²) / (3x² - 5xy)

5) f(x) = (6x² - 54) / (x² + 7x + 12)

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

7) f(x) = (x² + 7x + 10) / (x² - 4)

**Problem 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

**Problem 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

**Problem 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)

**Problem 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

**Problem 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)

**Problem 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

**Problem 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)

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