Question
Multiplication & Division of Rational Expressions Quick Check
5 of 55 of 5 Items
Question
Express in simplest form: 4x2−16x−848x3÷x2−9x2−3x .(1 point)
Responses
−72x
Start Fraction negative 7 over 2 x End Fraction
−45x−842x
Start Fraction negative 45 x minus 84 over 2 x End Fraction
2x2x−7
Start Fraction 2 x squared over x minus 7 End Fraction
x−72x2
Answers
GPT-4o mini
To simplify the expression \( \frac{4x^2 - 16x - 84}{8x^3} \div (x^2 - 9x^2 - 3x) \), we need to follow the steps:
1. **Simplify the numerator and the denominator separately**.
2. **Rewrite the division as multiplication by the reciprocal**.
3. **Factor out common terms and cancel them out**.
#### Step 1: Simplify the numerator
1. \( 4x^2 - 16x - 84 \)
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
2. \( 8x^3 \)
This term does not need simplifying.
3. \( x^2 - 9x^2 - 3x \)
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:
1. \(-72x\)
2. \(-\frac{45x - 84}{2x}\)
3. \( \frac{2x^2}{x - 7}\)
4. \(x - \frac{7}{2x^2}\)
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. **Simplify the numerator and the denominator separately**.
2. **Rewrite the division as multiplication by the reciprocal**.
3. **Factor out common terms and cancel them out**.
#### Step 1: Simplify the numerator
1. \( 4x^2 - 16x - 84 \)
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
2. \( 8x^3 \)
This term does not need simplifying.
3. \( x^2 - 9x^2 - 3x \)
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:
1. \(-72x\)
2. \(-\frac{45x - 84}{2x}\)
3. \( \frac{2x^2}{x - 7}\)
4. \(x - \frac{7}{2x^2}\)
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!