Sure, let's start by combining the like terms in the polynomial [tex]\(x y - 2 x y + 3 x^2 y - 4 x y^2 + 2 x y^2 - x y - 2 x y^2 + 3 x^2 y\)[/tex].
1. Identify and Combine Like Terms:
- Terms involving [tex]\(xy\)[/tex]: [tex]\(x y - 2 x y - x y\)[/tex]
- Terms involving [tex]\(x^2 y\)[/tex]: [tex]\(3 x^2 y + 3 x^2 y\)[/tex]
- Terms involving [tex]\(x y^2\)[/tex]: [tex]\(-4 x y^2 + 2 x y^2 - 2 x y^2\)[/tex]
Combining these like terms, we get:
- [tex]\(x y - 2 x y - x y = (1 - 2 - 1)xy = -2xy\)[/tex]
- [tex]\(3 x^2 y + 3 x^2 y = (3 + 3)x^2 y = 6x^2 y\)[/tex]
- [tex]\(-4 x y^2 + 2 x y^2 - 2 x y^2 = (-4 + 2 - 2)x y^2 = -4x y^2\)[/tex]
So the polynomial simplifies to:
[tex]\[
-2xy + 6x^2y - 4xy^2
\][/tex]
2. Evaluate the Polynomial for [tex]\(x = 4\)[/tex] and [tex]\(y = -2\)[/tex]:
[tex]\[
-2xy + 6x^2y - 4xy^2
\][/tex]
Substituting [tex]\(x = 4\)[/tex] and [tex]\(y = -2\)[/tex]:
[tex]\[
-2(4)(-2) + 6(4^2)(-2) - 4(4)(-2)^2
\][/tex]
3. Calculate Each Term:
- For [tex]\(-2xy\)[/tex]:
[tex]\[
-2(4)(-2) = 16
\][/tex]
- For [tex]\(6x^2y\)[/tex]:
[tex]\[
6(4^2)(-2) = 6(16)(-2) = 6 \times -32 = -192
\][/tex]
- For [tex]\(-4xy^2\)[/tex]:
[tex]\[
-4(4)(-2)^2 = -4(4)(4) = -4 \times 16 = -64
\][/tex]
4. Combine the Results:
[tex]\[
16 - 192 - 64 = 16 - 256 = -240
\][/tex]
Thus, the final evaluated value of the combined polynomial for [tex]\(x = 4\)[/tex] and [tex]\(y = -2\)[/tex] is:
[tex]\[
-240
\][/tex]
