SIMPLIFYING RATIONAL EXPRESSIONS
About "Simplifying rational expressions"
Simplifying rational expressions :
The expression which is in the form of f(x) / g(x) is called rational expression.
Simplifying rational expression is nothing but expressing the the rational expression in its lowest term or simplest form.
Simplifying rational expressions - Steps
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.
Simplifying rational expressions - Examples
Example 1 :
Write the simplified expression for the function defined by
f(x) = 4x³ / 8x²
Solution:
In this example we have x³ and x² . 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) = (4³a³ + 5³b³) / (4a²b + 5ab²)
f(x) = [(4a)³ + (5b)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)
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