Exercise 21 Page 690

We are given a rational expression and its simplified form. ccc Rational Expression & & Simplified Form [0.5em] p/x^2+x-12 & & x+5/x-3 Our goal is to find the expression that p represents. Since the original expression and its simplified version must be equal, we obtain the following equation. p/x^2+x-12=x+5/x-3 We can approach this equation in two ways.

1. Rewrite the left-hand side and conclude the form of p by analyzing the newly obtained equation.
2. Solve it algebraically, treating p as the variable.

   Let's take a look at each of these approaches, so you can choose which one you like better.

   First Approach

   To simplify any rational expression, we factor the numerator and the denominator, and cancel out their common factors. We are given the simplified form of a rational expression and the denominator of the original expression. We want to find the numerator of the original expression. To do so, we will perform the simplification in reverse. p/x^2+x-12=x+5/x-3 On the right-hand side of the equation, we can see that the remaining factor of the numerator is x+5 and the remaining factor of the denominator is x-3. The expression represented by p must consist of the remaining factor x+5 and the factor that was canceled out. ( x+5) ?/x^2+x-12=x+5/x-3 Let's factor the denominator of the original expression so that we can see all of its factors. This will help us identify the factor that was canceled out.
   x^2+x-12

   ▼

   Factor

   WriteDiff

   Write as a difference

   x^2+4x-3x-12

   FactorOut

   Factor out x

   x(x+4)-3x-12

   FactorOut

   Factor out - 3

   x(x+4)-3(x+4)

   FactorOut

   Factor out (x+4)

   (x+4)(x-3)
   Let's rewrite our equation using the factored form of the denominator. ( x+5) ?/x^2+x-12=x+5/x-3 ⇕ ( x+5) ?/( x+4)(x-3)=x+5/x-3 Looking at the above equation, we can see that the common factor of the numerator and the denominator that was canceled out is x+4. ( x+5) $( x+4)$/( x+4)(x-3)=x+5/x-3 We found the factored form of the numerator p of the original expression. Let's write our answer in standard form.
   p=( x+5)( x+4)

   ▼

   Simplify right-hand side

   Distr

   Distribute (x+4)

   p=x(x+4)+5(x+4)

   Distr

   Distribute x & 5

   p=x^2+4x+5x+20

   AddTerms

   Add terms

   p=x^2+9x+20

   Second Approach

   As mentioned before, we can also find the expression represented by p by solving the equation. p/x^2+x-12=x+5/x-3 Remember we will treat p as the variable. Therefore, to solve the equation, we will isolate p on the left-hand side of the equation.
   p/x^2+x-12=x+5/x-3

   ▼

   Solve for p

   MultEqn

   LHS * (x^2+x-12)=RHS* (x^2+x-12)

   p/x^2+x-12*(x^2+x-12)=x+5/x-3*(x^2+x-12)

   FracMultDenomToNumber

   a/(x^2+x-12)* (x^2+x-12) = a

   p=x+5/x-3*(x^2+x-12)

   MoveRightFacToNum

   a/c* b = a* b/c

   p=(x+5)(x^2+x-12)/x-3
   Now that p is isolated, we can simplify the right-hand side of the equation by first factoring the trinomial and canceling any common factors. Then, we can rewrite the remaining expression in standard form.
   p=(x+5)(x^2+x-12)/x-3

   ▼

   Factor

   WriteDiff

   Write as a difference

   p=(x+5)(x^2+4x-3x-12)/x-3

   FactorOut

   Factor out x & - 3

   p=(x+5)(x(x+4)-3(x+4))/x-3

   FactorOut

   Factor out (x+4)

   p=(x+5)(x+4)(x-3)/x-3

   CancelCommonFac

   Cancel out common factors

   p=(x+5)(x+4)(x-3)/x-3

   SimpQuot

   Simplify quotient

   p=(x+5)(x+4)

   ▼

   Simplify right-hand side

   Distr

   Distribute (x+4)

   p=x(x+4)+5(x+4)

   Distr

   Distribute x & 5

   p=x^2+4x+5x+20

   AddTerms

   Add terms

   p=x^2+9x+20

   Conclusion

   We found that p=x^2+9x+20 using two different methods of solving our equation. Both of these methods are equally valid, so it is up to you to choose which one you prefer.