To find the difference between the two polynomial expressions:
[tex]\[
(4 x^2 y^3 + 2 x y^2 - 2 y) - (-7 x^2 y^3 + 6 x y^2 - 2 y),
\][/tex]
we need to subtract the corresponding coefficients of [tex]\(x^2 y^3\)[/tex], [tex]\(x y^2\)[/tex], and [tex]\(y\)[/tex] from each expression.
1. Identify and arrange the coefficients:
For the first expression [tex]\(4 x^2 y^3 + 2 x y^2 - 2 y\)[/tex]:
- The coefficient of [tex]\(x^2 y^3\)[/tex] is 4.
- The coefficient of [tex]\(x y^2\)[/tex] is 2.
- The coefficient of [tex]\(y\)[/tex] is -2.
For the second expression [tex]\(-7 x^2 y^3 + 6 x y^2 - 2 y\)[/tex]:
- The coefficient of [tex]\(x^2 y^3\)[/tex] is -7.
- The coefficient of [tex]\(x y^2\)[/tex] is 6.
- The coefficient of [tex]\(y\)[/tex] is -2.
2. Subtract the coefficients of the corresponding terms:
[tex]\[
\text{Coefficient of } x^2 y^3: 4 - (-7) = 4 + 7 = 11
\][/tex]
[tex]\[
\text{Coefficient of } x y^2: 2 - 6 = -4
\][/tex]
[tex]\[
\text{Coefficient of } y: -2 - (-2) = -2 + 2 = 0
\][/tex]
3. Place the coefficients in the resulting polynomial:
Thus, the difference is:
[tex]\[
11 x^2 y^3 - 4 x y^2 + 0 y
\][/tex]
Since the coefficient of [tex]\(y\)[/tex] is 0, it does not contribute to the polynomial, so it can be omitted.
Therefore, the polynomial difference is:
[tex]\[
11 x^2 y^3 - 4 x y^2
\][/tex]
So, the correct coefficients in the polynomial difference are:
[tex]\[
\boxed{11} x^2 y^3 + \boxed{-4} x y^2 + \quad \boxed{0} y
\][/tex]
