Let's solve the given mathematical expression step by step.
We start with the given problem:
[tex]$ (x^3 + 2x^2 - 9x - 18) \div \frac{x^2 + 11x + 24}{x^2 - 11x + 24} \div \frac{x^2 - 6x - 16}{x^2 + 5x - 24} $[/tex]
---
Step 1: Simplify the first division
Rewrite the first division as a multiplication by the reciprocal:
[tex]$ (x^3 + 2x^2 - 9x - 18) \div \frac{x^2 + 11x + 24}{x^2 - 11x + 24} = (x^3 + 2x^2 - 9x - 18) \cdot \frac{x^2 - 11x + 24}{x^2 + 11x + 24} $[/tex]
---
Step 2: Combine the divisions
First, combine the two divisions into one expression by using reciprocals:
[tex]$ (x^3 + 2x^2 - 9x - 18) \cdot \frac{x^2 - 11x + 24}{x^2 + 11x + 24} \div \frac{x^2 - 6x - 16}{x^2 + 5x - 24} = (x^3 + 2x^2 - 9x - 18) \cdot \frac{x^2 - 11x + 24}{x^2 + 11x + 24} \cdot \frac{x^2 + 5x - 24}{x^2 - 6x - 16} $[/tex]
---
Step 3: Simplify each polynomial fraction
Factor each polynomial:
1. [tex]\( x^3 + 2x^2 - 9x - 18 = (x - 3)(x + 3)(x + 2) \)[/tex]
2. [tex]\( x^2 + 11x + 24 = (x + 3)(x + 8) \)[/tex]
3. [tex]\( x^2 - 11x + 24 = (x - 3)(x - 8) \)[/tex]
4. [tex]\( x^2 + 5x - 24 = (x + 8)(x - 3) \)[/tex]
5. [tex]\( x^2 - 6x - 16 = (x - 8)(x + 2) \)[/tex]
Substitute the factored forms in:
[tex]$ \left( (x - 3)(x + 3)(x + 2) \right) \cdot \frac{(x - 3)(x - 8)}{(x + 3)(x + 8)} \cdot \frac{(x + 8)(x - 3)}{(x - 8)(x + 2)} $[/tex]
---
Step 4: Simplify the expression
Cancel out the common factors in the numerator and the denominator:
[tex]$ = (x - 3) \cdot \frac{(x - 3)}{(x + 8)} \cdot \frac{(x + 8)}{(x - 8)} $[/tex]
[tex]$ = (x - 3)^3 $[/tex]
This simplifies to:
[tex]$ x^3 - 9x^2 + 27x - 27 $[/tex]
Thus, our simplified solution is:
[tex]$ x^3 - 9x^2 + 27x - 27 $[/tex]
This is the simplified form of the given division problem.
