To divide the given expression
[tex]\[
\frac{16 x^2 y^2 + 8 x y^2 - 16 y^2}{4 x^2 y}
\][/tex]
we will split it into three separate fractions, each corresponding to a term in the numerator. Let's perform the division step by step for each term:
1. First term: [tex]\( \frac{16 x^2 y^2}{4 x^2 y} \)[/tex]
- Simplify the coefficients: [tex]\( \frac{16}{4} = 4 \)[/tex]
- Simplify the [tex]\( x \)[/tex]-terms: [tex]\( \frac{x^2}{x^2} = 1 \)[/tex]
- Simplify the [tex]\( y \)[/tex]-terms: [tex]\( \frac{y^2}{y} = y \)[/tex]
Thus, the first term simplifies to:
[tex]\[
4y
\][/tex]
2. Second term: [tex]\( \frac{8 x y^2}{4 x^2 y} \)[/tex]
- Simplify the coefficients: [tex]\( \frac{8}{4} = 2 \)[/tex]
- Simplify the [tex]\( x \)[/tex]-terms: [tex]\( \frac{x}{x^2} = \frac{1}{x} \)[/tex]
- Simplify the [tex]\( y \)[/tex]-terms: [tex]\( \frac{y^2}{y} = y \)[/tex]
Thus, the second term simplifies to:
[tex]\[
\frac{2y}{x}
\][/tex]
3. Third term: [tex]\( \frac{-16 y^2}{4 x^2 y} \)[/tex]
- Simplify the coefficients: [tex]\( \frac{-16}{4} = -4 \)[/tex]
- Simplify the [tex]\( x \)[/tex]-terms: there is no [tex]\( x \)[/tex]-term in the numerator, so it remains [tex]\( \frac{1}{x^2} \)[/tex]
- Simplify the [tex]\( y \)[/tex]-terms: [tex]\( \frac{y^2}{y} = y \)[/tex]
Thus, the third term simplifies to:
[tex]\[
\frac{-4y}{x^2}
\][/tex]
Finally, combining all these simplified terms, we get:
[tex]\[
4y + \frac{2y}{x} - \frac{4y}{x^2}
\][/tex]
The coefficients of these terms are [tex]\( 4, 2, \)[/tex] and [tex]\( -4 \)[/tex], respectively.
