HOW TO FIND THE EXCLUDED VALUE OF A RATIONAL EXPRESSION

Example 1 :

Find the excluded values of the following expression, if any.

y/(y² - 25)

Solution :

=  y/(y² - 25)

y² - 25  =  y² - 5²

  =  (y - 5)(y + 5)

=  y/(y + 5) (y - 5)

By equating the denominator equal to zero, we get y + 5 = 0 and y - 5  =  0.

y + 5 = 0  and  y - 5  =  0

 y = -5 and y = 5

So, the excluded values are -5 and 5.

Example 2 :

Find the excluded values of the following expression, if any.

t/(t² - 5t + 6)  

Solution :

=  t/(t² - 5t + 6)  

 t² - 5t + 6  =  (t - 2)(t - 3)

=  t/(t - 2)(t - 3)

By equating the denominator equal to zero, we get t- 2  = 0 and t - 3  =  0

t - 2 = 0  and  t - 3  =  0

 y = 2 and y = 3

So, the excluded values are 2 and 3.

Example 3 :

Find the excluded values of the following expression, if any.

(x² + 6x + 8)/(x² + x - 2)

Solution :

=  (x² + 6x + 8)/(x² + x - 2) 

x² + 6x + 8  =  (x + 2)(x + 4)

(x² + x - 2)  =  (x + 2)(x - 1)

=  (x + 2)(x + 4)/(x + 2)(x - 1)

=  (x + 4)/(x - 1)

By equating the denominator equal to zero, we get x - 1  =  0

So, the excluded value is 1.

Example 4 :

Find the excluded values of the following expression, if any.

(x³ - 27) / (x³ + x² - 6x)

Solution :

x³ - 27  =  x³ - 3³ 

  =   (x - 3)(x² + x(3) + 3²)

  =   (x - 3)(x² + 3x + 9)

x³ + x² - 6x  =  x(x² + x - 6)

 =  x(x + 3)(x - 2)

 =  (x - 3)(x² - 3x + 9) / x(x + 3)(x - 2)

By equating the denominator equal to zero, we get x = 0, x + 3 = 0 and x - 2 = 0

So, the excluded value is 0, 2 and 3.

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