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 subtract the corresponding terms from each expression step-by-step.
1. Identify the terms in each polynomial:
For the first expression [tex]\(4 x^2 y^3 + 2 x y^2 - 2 y\)[/tex]:
- Term involving [tex]\(x^2 y^3\)[/tex]: [tex]\(4 x^2 y^3\)[/tex]
- Term involving [tex]\(x y^2\)[/tex]: [tex]\(2 x y^2\)[/tex]
- Term involving [tex]\(y\)[/tex]: [tex]\(-2 y\)[/tex]
For the second expression [tex]\(-7 x^2 y^3 + 6 x y^2 - 2 y\)[/tex]:
- Term involving [tex]\(x^2 y^3\)[/tex]: [tex]\(-7 x^2 y^3\)[/tex]
- Term involving [tex]\(x y^2\)[/tex]: [tex]\(6 x y^2\)[/tex]
- Term involving [tex]\(y\)[/tex]: [tex]\(-2 y\)[/tex]
2. Subtract the corresponding terms:
- For the term involving [tex]\(x^2 y^3\)[/tex]:
[tex]\(4 x^2 y^3 - (-7 x^2 y^3) = 4 x^2 y^3 + 7 x^2 y^3 = 11 x^2 y^3\)[/tex]
- For the term involving [tex]\(x y^2\)[/tex]:
[tex]\(2 x y^2 - 6 x y^2 = 2 x y^2 - 6 x y^2 = -4 x y^2\)[/tex]
- For the term involving [tex]\(y\)[/tex]:
[tex]\(-2 y - (-2 y) = -2 y + 2 y = 0\)[/tex]
3. Combine the results to form the final expression:
- The term involving [tex]\(x^2 y^3\)[/tex] has a coefficient of 11
- The term involving [tex]\(x y^2\)[/tex] has a coefficient of -4
- The term involving [tex]\(y\)[/tex] has a coefficient of 0
Therefore, the final expression is:
[tex]\[
11 x^2 y^3 - 4 x y^2 + 0 y \implies 11 x^2 y^3 - 4 x y^2
\][/tex]
Hence, placing the correct coefficients in the difference:
- The coefficient before [tex]\(x^2 y^3\)[/tex] is [tex]\(11\)[/tex]
- The coefficient before [tex]\(x y^2\)[/tex] is [tex]\(-4\)[/tex]
- The coefficient before [tex]\(y\)[/tex] is [tex]\(0\)[/tex]
So, the coefficients are:
[tex]\[
11, -4 x y^2, 0 y
\][/tex]
In the final simplified form, we just have:
[tex]\[
11x^2y^3 - 4xy^2
\][/tex]
