Simplify the following rational expression. List all of the excluded values, and classify each as a removable discontinuity, or a nonremovable discontinuity. -2/x^2-4+x-1/x^2-2x

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Excluded values  -2,0,2  These are non removable since we cannot cancel them in the numerator

(x+1)

x(x+2)  

Step-by-step explanation:

-2                  x-1

+

x^2 -4        x^2 -2x

First factor the denominators

-2                  x-1

+

x^2 -4        x^2 -2x

x^2 -4 is the difference of squares

x^2 -4 = (x-2) (x+2)

x^2 -2x has an x in common

x(x-2)

-2                   x-1

+    

(x-2)(x+2)        x(x -2)

The values that are excluded are the values that make the denominators zero

(x-2) (x+2) = 0     x=2  x=-2

x(x-2) =0  x=0  x=2

Excluded values  -2,0,2  These are non removable since we cannot cancel them in the numerator

We need to get a common denominator of (x-2)(x+2)x

Multiply the first term by x/x and the second term by (x+2)/(x+2)

-2 x                 (x-1) (x+2)

+    

x(x-2)(x+2)        x(x -2)(x+2)

Distribute

(x+2)(x-1) = x^2 -x+2x-2 = x^2+x-2

-2 x                x^2+x-2

+    

x(x-2)(x+2)        x(x -2)(x+2)

Combine like terms, since the denominators are the same

-2 x  + x^2+x-2

x(x-2)(x+2)    

 x^2-x-2

x(x-2)(x+2)      

Factor the numerator

 x^2-x-2 = (x-2) (x+1)

(x-2) (x+1)

x(x-2)(x+2)  

Cancel like terms

(x+1)

x(x+2)  

The excluded values are found at the beginning, not now.  We canceled some of them in our simplified expression.  They are still excluded values

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step-by-step explanation:

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