Sure, let's combine the like terms in the given polynomial and then evaluate it for [tex]\( x = 3 \)[/tex] and [tex]\( y = -1 \)[/tex].
The polynomial given is:
[tex]\[ x y + 3 x y^2 - 2 x y + 3 x^2 y - 2 x y^2 + x y \][/tex]
First, let's identify and group the like terms:
- Terms containing [tex]\( xy \)[/tex]: [tex]\( x y, -2 x y, x y \)[/tex]
- Terms containing [tex]\( xy^2 \)[/tex]: [tex]\( 3 x y^2, -2 x y^2 \)[/tex]
- Terms containing [tex]\( x^2y \)[/tex]: [tex]\( 3 x^2 y \)[/tex]
Now, let's combine these like terms:
1. For the [tex]\( xy \)[/tex] terms:
[tex]\[ x y - 2 x y + x y = (1 - 2 + 1) x y = 0 x y \][/tex]
2. For the [tex]\( xy^2 \)[/tex] terms:
[tex]\[ 3 x y^2 - 2 x y^2 = (3 - 2) x y^2 = 1 x y^2 \][/tex]
3. For the [tex]\( x^2y \)[/tex] term:
[tex]\[ 3 x^2 y \][/tex]
So, combining all like terms, we have:
[tex]\[ 0 x y + x y^2 + 3 x^2 y = x y^2 + 3 x^2 y \][/tex]
Now we need to evaluate this simplified polynomial at [tex]\( x = 3 \)[/tex] and [tex]\( y = -1 \)[/tex]:
[tex]\[ x y^2 + 3 x^2 y \][/tex]
Substitute [tex]\( x = 3 \)[/tex] and [tex]\( y = -1 \)[/tex]:
[tex]\[ 3(-1)^2 + 3(3)^2(-1) \][/tex]
Calculate the value step-by-step:
1. Evaluate [tex]\( (-1)^2 = 1 \)[/tex],
[tex]\[ 3(1) + 3(3)^2(-1) \][/tex]
2. Evaluate [tex]\( (3)^2 = 9 \)[/tex],
[tex]\[ 3(1) + 3(9)(-1) \][/tex]
3. Then,
[tex]\[ 3 + 3(9)(-1) = 3 + 3*(-9) \][/tex]
4. Finally,
[tex]\[ 3 + (-27) = -24 \][/tex]
Thus, the simplified polynomial is [tex]\( x y^2 + 3 x^2 y \)[/tex], and its value when [tex]\( x = 3 \)[/tex] and [tex]\( y = -1 \)[/tex] is [tex]\(-24\)[/tex].
So, the correct option in your list is:
[tex]\[ x y^2 + 3 x^2 y ;-24 \][/tex]
