Find: [tex]\left(4x^2 y^3 + 2xy^2 - 2y\right) - \left(-7x^2 y^3 + 6xy^2 - 2y\right)[/tex]

Place the correct coefficients in the difference.

[tex]\square x^2 y^3 + \square xy^2 + \square y[/tex]



Answer :

To solve the expression [tex]\(\left(4 x^2 y^3 + 2 x y^2 - 2 y\right) - \left(-7 x^2 y^3 + 6 x y^2 - 2 y\right)\)[/tex], we need to follow these steps:

1. Distribute the negative sign through the second polynomial:

[tex]\[ - \left(-7 x^2 y^3 + 6 x y^2 - 2 y\right) = 7 x^2 y^3 - 6 x y^2 + 2 y \][/tex]

2. Combine like terms from the two polynomials:

[tex]\[ (4 x^2 y^3 + 2 x y^2 - 2 y) + (7 x^2 y^3 - 6 x y^2 + 2 y) \][/tex]

3. Add the coefficients of [tex]\(x^2 y^3\)[/tex]:

[tex]\[ 4 x^2 y^3 + 7 x^2 y^3 = (4 + 7)x^2 y^3 = 11 x^2 y^3 \][/tex]

4. Add the coefficients of [tex]\(x y^2\)[/tex]:

[tex]\[ 2 x y^2 - 6 x y^2 = (2 - 6)x y^2 = -4 x y^2 \][/tex]

5. Add the coefficients of [tex]\(y\)[/tex]:

[tex]\[ -2 y + 2 y = (-2 + 2)y = 0 y \][/tex]

6. Combine the results:

[tex]\[ 11 x^2 y^3 - 4 x y^2 + 0 y \][/tex]

Thus, the coefficients in the difference are:
[tex]\[ 11 \quad x^2 y^3 \quad + \quad (-4) \quad x y^2 \quad + \quad 0 \quad y \][/tex]

So, the final answer is:
[tex]\[ 11 \ x^2 y^3 + (-4) \ x y^2 + 0 \ y \][/tex]

Or more simply:

[tex]\[ 11 x^2 y^3 - 4 x y^2 \][/tex]

The correct coefficients for the difference are:

[tex]\[ \boxed{11} \ x^2 y^3 + \boxed{-4} \ x y^2 + \boxed{0} \ y \][/tex]