Questions: Divide the polynomial by the monomial denominator by writing the fraction as the sum (or difference) of fractions [ (3 x^5+9 x^3-4)/(x^4) ]

Solution Steps

To divide the polynomial by the monomial, we will split the given polynomial into separate fractions, each with the monomial as the denominator. Then, we will simplify each fraction individually.

We start with the polynomial \( 3x^5 + 9x^3 - 4 \) and the monomial denominator \( x^4 \). To divide the polynomial by the monomial, we can express the division as separate fractions:

\[ \frac{3x^5}{x^4} + \frac{9x^3}{x^4} - \frac{4}{x^4} \]

Now, we simplify each of the fractions:

1. For \( \frac{3x^5}{x^4} \): \[ \frac{3x^5}{x^4} = 3x^{5-4} = 3x \]

2. For \( \frac{9x^3}{x^4} \): \[ \frac{9x^3}{x^4} = 9x^{3-4} = \frac{9}{x} \]

3. For \( \frac{4}{x^4} \): \[ \frac{4}{x^4} = \frac{4}{x^4} \]

Combining the simplified fractions, we have:

\[ 3x + \frac{9}{x} - \frac{4}{x^4} \]

Final Answer