Warner Owen
06/15/2024 · High School
Perform the division. \[ \frac{3 x^{4}-2 x^{3}+7}{-3 x^{4}} \]
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Step-by-step Solution
To perform the division of the polynomial \( \frac{3 x^{4}-2 x^{3}+7}{-3 x^{4}} \), we can separate the terms in the numerator and divide each by the denominator.
Starting with the expression:
\[
\frac{3 x^{4}-2 x^{3}+7}{-3 x^{4}} = \frac{3 x^{4}}{-3 x^{4}} + \frac{-2 x^{3}}{-3 x^{4}} + \frac{7}{-3 x^{4}}
\]
Now, we will simplify each term individually:
1. For the first term:
\[
\frac{3 x^{4}}{-3 x^{4}} = -1
\]
2. For the second term:
\[
\frac{-2 x^{3}}{-3 x^{4}} = \frac{2}{3} \cdot \frac{x^{3}}{x^{4}} = \frac{2}{3} \cdot \frac{1}{x} = \frac{2}{3x}
\]
3. For the third term:
\[
\frac{7}{-3 x^{4}} = -\frac{7}{3 x^{4}}
\]
Now, we can combine all the simplified terms:
\[
-1 + \frac{2}{3x} - \frac{7}{3x^{4}}
\]
Thus, the final result of the division is:
\[
\boxed{-1 + \frac{2}{3x} - \frac{7}{3x^{4}}}
\]
Quick Answer
The result of the division is \(-1 + \frac{2}{3x} - \frac{7}{3x^{4}}\).
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