Simplify (z+3)/(z^2+3z+2)-(z-1)/(z^2+z-2)

Problem

(z+3)/(z2+3*z+2)−(z−1)/(z2+z−2)

Solution

1. Factor the denominators of both rational expressions.

z2+3*z+2=(z+1)*(z+2)

z2+z−2=(z+2)*(z−1)

2. Rewrite the expression using the factored denominators.

(z+3)/((z+1)*(z+2))−(z−1)/((z+2)*(z−1))

3. Simplify the second fraction by canceling the common factor (z−1) noting that z≠1

(z+3)/((z+1)*(z+2))−1/(z+2)

4. Find a common denominator for the two fractions, which is (z+1)*(z+2)

(z+3)/((z+1)*(z+2))−(1*(z+1))/((z+2)*(z+1))

5. Subtract the numerators over the common denominator.

(z+3−(z+1))/((z+1)*(z+2))

6. Distribute the negative sign and combine like terms in the numerator.

(z+3−z−1)/((z+1)*(z+2))

2/((z+1)*(z+2))

Final Answer

(z+3)/(z2+3*z+2)−(z−1)/(z2+z−2)=2/((z+1)*(z+2))

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