- Convert the division of the rational expression into multiplication by the reciprocal: \(\frac{4x^2 - 16x - 84}{8x^3} \cdot \frac{x^2 - 3x}{x^2 - 9}\).
- Factor each polynomial: \(4x^2 - 16x - 84 = 4(x-7)(x+3)\), \(x^2 - 3x = x(x-3)\), and \(x^2 - 9 = (x-3)(x+3)\).
- Substitute the factored forms and cancel common terms: \(\frac{4(x-7)(x+3)}{8x^3} \cdot \frac{x(x-3)}{(x-3)(x+3)}\).
- Simplify the remaining terms to reach the final form \(\boxed{\frac{x-7}{2x^2}}\).
Explanation
- Converting to Multiplication and Factoring Strategy
To simplify the given rational expression \(\frac{4x^2 - 16x - 84}{8x^3} \div \frac{x^2 - 9}{x^2 - 3x}\), we first recall that dividing by a fraction is equivalent to multiplying by its reciprocal. Thus, we rewrite the expression as:
\[ \frac{4x^2 - 16x - 84}{8x^3} \cdot \frac{x^2 - 3x}{x^2 - 9} \]
We will now factor each numerator and denominator individually to simplify the expression.
- Factoring Numerators and Denominators
First, we factor the numerator \(4x^2 - 16x - 84\):
\[ 4(x^2 - 4x - 21) \]
Finding two numbers that multiply to \(-21\) and add to \(-4\), we get \(-7\) and \(3\). So, the numerator factors as:
\[ 4(x - 7)(x + 3) \]
Next, we factor the expression \(x^2 - 3x\) in the second numerator:
\[ x(x - 3) \]
Then, we factor the denominator \(x^2 - 9\) in the second fraction as a difference of squares:
\[ (x - 3)(x + 3) \]
Finally, the denominator \(8x^3\) remains as is for now.
- Canceling Common Factors and Final Simplification
Now, we substitute the factored forms back into our expression:
\[ \frac{4(x - 7)(x + 3)}{8x^3} \cdot \frac{x(x - 3)}{(x - 3)(x + 3)} \]
We identify common factors in the numerators and denominators that can be canceled:
- The factor \((x + 3)\) appears in the first numerator and the second denominator.
- The factor \((x - 3)\) appears in the second numerator and the second denominator.
- The expression \(\frac{4}{8}\) simplifies to \(\frac{1}{2}\).
- The term \(x\) in the second numerator and \(x^3\) in the first denominator simplifies to \(\frac{1}{x^2}\).
Canceling these factors, we are left with:
\[ \frac{1 \cdot (x - 7) \cdot 1}{2 \cdot x^2 \cdot 1} = \frac{x - 7}{2x^2} \]
- Conclusion
Following our systematic factorization and cancellation, the original expression simplifies to \(\frac{x-7}{2x^2}\). Comparing this to the given options, this matches the first option.
The final answer is \(\boxed{\frac{x-7}{2x^2}}\).
Examples
Rational expression simplification is essential in fields like physics and engineering, where complex formulas are reduced to manageable forms to analyze relationships between variables. For example, if you are modeling the velocity of a particle with a complicated rational function, simplifying the expression can help you quickly determine its acceleration or find the points where the velocity is zero, making calculations for complex systems far more efficient.