To find the difference between the two polynomials [tex]\((-2 x^3 y^2 + 4 x^2 y^3 - 3 x y^4)\)[/tex] and [tex]\((6 x^4 y - 5 x^2 y^3 - y^5)\)[/tex], follow these steps:
1. Write both polynomials separately:
- The first polynomial is [tex]\(P_1 = -2 x^3 y^2 + 4 x^2 y^3 - 3 x y^4\)[/tex].
- The second polynomial is [tex]\(P_2 = 6 x^4 y - 5 x^2 y^3 - y^5\)[/tex].
2. Subtract the second polynomial from the first polynomial:
[tex]\[
P_1 - P_2 = \left(-2 x^3 y^2 + 4 x^2 y^3 - 3 x y^4\right) - \left(6 x^4 y - 5 x^2 y^3 - y^5\right)
\][/tex]
3. Distribute the subtraction across the second polynomial:
[tex]\[
P_1 - P_2 = -2 x^3 y^2 + 4 x^2 y^3 - 3 x y^4 - 6 x^4 y + 5 x^2 y^3 + y^5
\][/tex]
4. Combine like terms:
- The term with [tex]\(x^4 y\)[/tex]: [tex]\(0 - 6 x^4 y = -6 x^4 y\)[/tex].
- The term with [tex]\(x^3 y^2\)[/tex]: [tex]\(-2 x^3 y^2 + 0 = -2 x^3 y^2\)[/tex].
- The term with [tex]\(x^2 y^3\)[/tex]: [tex]\(4 x^2 y^3 + 5 x^2 y^3 = 9 x^2 y^3\)[/tex].
- The term with [tex]\(x y^4\)[/tex]: [tex]\(-3 x y^4 + 0 = -3 x y^4\)[/tex].
- The term with [tex]\(y^5\)[/tex]: [tex]\(0 + y^5 = y^5\)[/tex].
5. Write the simplified polynomial:
[tex]\[
P_1 - P_2 = -6 x^4 y - 2 x^3 y^2 + 9 x^2 y^3 - 3 x y^4 + y^5
\][/tex]
So, the difference of the polynomials is:
[tex]\[
y(-6 x^4 - 2 x^3 y + 9 x^2 y^2 - 3 x y^3 - y^4)
\][/tex]
The correct answer is [tex]\(-6 x^4 y - 2 x^3 y^2 + 9 x^2 y^3 - 3 x y^4 + y^5\)[/tex]. This matches with the first option provided.
