Sure, let's simplify the given polynomial step by step and then evaluate it for [tex]\( x = -2 \)[/tex] and [tex]\( y = -3 \)[/tex].
1. Combine like terms in the polynomial:
The given polynomial is:
[tex]\[
2xy^2 + 3x - 4y^2 + 2xy^2 - 3xy^2 - x
\][/tex]
Let's group the like terms:
[tex]\[
(2xy^2 + 2xy^2 - 3xy^2) + (3x - x) - 4y^2
\][/tex]
Combine the [tex]\( xy^2 \)[/tex] terms:
[tex]\[
(2 + 2 - 3)xy^2 + (3 - 1)x - 4y^2
\][/tex]
[tex]\[
xy^2 + 2x - 4y^2
\][/tex]
So, the simplified polynomial is:
[tex]\[
xy^2 + 2x - 4y^2
\][/tex]
2. Evaluate the simplified polynomial for [tex]\( x = -2 \)[/tex] and [tex]\( y = -3 \)[/tex]:
Let's substitute [tex]\( x = -2 \)[/tex] and [tex]\( y = -3 \)[/tex] into the simplified polynomial:
[tex]\[
(-2)(-3)^2 + 2(-2) - 4(-3)^2
\][/tex]
3. Compute each term separately:
- For the term [tex]\( xy^2 \)[/tex]:
[tex]\[
(-2)(-3)^2 = (-2)(9) = -18
\][/tex]
- For the term [tex]\( 2x \)[/tex]:
[tex]\[
2(-2) = -4
\][/tex]
- For the term [tex]\( -4y^2 \)[/tex]:
[tex]\[
-4(-3)^2 = -4(9) = -36
\][/tex]
4. Add the computed values:
[tex]\[
-18 + (-4) + (-36)
\][/tex]
Combine the values:
[tex]\[
-18 - 4 - 36 = -58
\][/tex]
So, the value of the polynomial [tex]\( 2xy^2 + 3x - 4y^2 + 2xy^2 - 3xy^2 - x \)[/tex] when evaluated for [tex]\( x = -2 \)[/tex] and [tex]\( y = -3 \)[/tex] is [tex]\(-58\)[/tex]. Additionally, the individual contributions to the result are [tex]\(-18\)[/tex], [tex]\(-4\)[/tex], and [tex]\(-36\)[/tex] respectively.
