To divide the polynomial [tex]\( \frac{66 x^3 y^2 - 110 x^2 y^3 - 44 x^4 y^3}{11 x^2 y^2} \)[/tex], we can break it down step-by-step, simplifying each term in the numerator by the monomial in the denominator individually.
The given polynomial is:
[tex]\[ \frac{66 x^3 y^2 - 110 x^2 y^3 - 44 x^4 y^3}{11 x^2 y^2} \][/tex]
Let's divide each term in the numerator separately:
### 1. Simplify the first term:
[tex]\[ \frac{66 x^3 y^2}{11 x^2 y^2} \][/tex]
- Divide the coefficients:
[tex]\[ \frac{66}{11} = 6 \][/tex]
- Subtract the exponents of [tex]\( x \)[/tex]:
[tex]\[ x^{3-2} = x \][/tex]
- Subtract the exponents of [tex]\( y \)[/tex]:
[tex]\[ y^{2-2} = y^0 = 1 \][/tex]
So the first term simplifies to:
[tex]\[ 6x \][/tex]
### 2. Simplify the second term:
[tex]\[ \frac{-110 x^2 y^3}{11 x^2 y^2} \][/tex]
- Divide the coefficients:
[tex]\[ \frac{-110}{11} = -10 \][/tex]
- Subtract the exponents of [tex]\( x \)[/tex]:
[tex]\[ x^{2-2} = x^0 = 1 \][/tex]
- Subtract the exponents of [tex]\( y \)[/tex]:
[tex]\[ y^{3-2} = y \][/tex]
So the second term simplifies to:
[tex]\[ -10y \][/tex]
### 3. Simplify the third term:
[tex]\[ \frac{-44 x^4 y^3}{11 x^2 y^2} \][/tex]
- Divide the coefficients:
[tex]\[ \frac{-44}{11} = -4 \][/tex]
- Subtract the exponents of [tex]\( x \)[/tex]:
[tex]\[ x^{4-2} = x^2 \][/tex]
- Subtract the exponents of [tex]\( y \)[/tex]:
[tex]\[ y^{3-2} = y \][/tex]
So the third term simplifies to:
[tex]\[ -4x^2 y \][/tex]
### Putting it all together:
Combining our simplified terms, we get:
[tex]\[ 6x - 10y - 4x^2 y \][/tex]
Thus, the simplified polynomial is:
[tex]\[ \boxed{6x - 10y - 4x^2 y} \][/tex]
