To simplify the expression \( \frac{4x^2 - 16x - 84}{8x^3} \div (x^2 - 9x^2 - 3x) \), we need to follow the steps:
- Simplify the numerator and the denominator separately.
- Rewrite the division as multiplication by the reciprocal.
- Factor out common terms and cancel them out.
Step 1: Simplify the numerator
- \( 4x^2 - 16x - 84 \)
To factor this, we look for numbers that multiply to \(4*(-84) = -336\) and add to \(-16\). These numbers are \(-28\) and \(12\).
Thus, \[ 4x^2 - 16x - 84 = 4(x^2 - 4x - 21) = 4(x - 7)(x + 3) \]
Step 2: Simplify the denominator
- \( 8x^3 \)
This term does not need simplifying.
- \( x^2 - 9x^2 - 3x \)
Note \( x^2 - 9x^2 = -8x^2 \), \[ -x^2 - 3x = -x(x + 3) \]
Thus, \[ x^2 - 9x^2 - 3x = -x(x + 3) = -x^2 - 3x \]
Step 3: Rewrite the division as multiplication
Now let's set up our rewritten expression: \[ \frac{4(x - 7)(x + 3)}{8x^3} \div (-x(x + 3))} \] This becomes: \[ \frac{4(x - 7)(x + 3)}{8x^3} \cdot \left(-\frac{1}{x(x + 3)}\right) \]
Step 4: Combine the fractions and simplify
\[ = \frac{4(x - 7)(x + 3)(-1)}{8x^3(x(x + 3))} = \frac{-4(x - 7)}{8x^4} \]
Now we can simplify further. The \(4\) in the numerator and the \(8\) in the denominator can be simplified: \[ \frac{-4}{8} = -\frac{1}{2} \]
So we have: \[ \frac{-(x - 7)}{2x^4} = \frac{-(x - 7)}{2x^4} = \frac{-x + 7}{2x^4} \]
Final Expression:
This results in: \[ \frac{7 - x}{2x^4} \]
None of the answers provided exactly matches this result, but it is incorrect to assume the input for the mathematical representation leads to a simple choice.
If you can provide a check with the answer options given:
- \(-72x\)
- \(-\frac{45x - 84}{2x}\)
- \( \frac{2x^2}{x - 7}\)
- \(x - \frac{7}{2x^2}\)
Among the provided options, there seems to be the intended closest options for further inspection or verifying based on standard outputs.
The completing of this final summary:
- The significant reductions, leads us to yield \(-\frac{1/2(x-7 )}{x^4}\).
Of course, simplifications can vary based on the algebraic paths one engages and further analysis.
Let me know how have I displayed processes further align to optimally highlight the answers prescribed. Thank you!
1 answer