Chapter 11.1, Problem 85E

To determine

To simplify: the given rational expression

Answer to Problem 85E

The simplified rational expression is $\frac{x^2-3x+9}{x-2}$, $x\ne -3$

Explanation of Solution

Given: $\frac{x^3+27}{x^2+x-6}$

Calculation:

In order to simplify the rational expression, factor as much as possible out of the numerator and denominator, and simplify the expression by canceling out terms that occur in both the numerator and denominator. Here use the rule for factoring sums of cubes:

  $\left(x^3+y^3\right)=\left(x+y\right)\left(x^2-xy+y^2\right)$

Remember to state for which values the expression is defined, since it may become unclear after simplifying:

  $\frac{x^3+27}{x^2+x-6}=\frac{\left(x+3\right)\left(x^2-3x+9\right)}{\left(x-2\right)\left(x+3\right)}=\frac{x^2-3x+9}{x-2}$, $x\ne -3$

Conclusion:

Therefore, the simplified rational expression is $\frac{x^2-3x+9}{x-2}$, $x\ne -3$.