Let's add the given polynomials step by step.
We have two polynomials:
[tex]\[ P(x, y) = -7 + 4x^2 + 3xy^2 \][/tex]
[tex]\[ Q(x, y) = 6x^3 - x^2 + xy^2 \][/tex]
To add these polynomials, we combine like terms. Like terms are terms that have the same variables raised to the same powers.
First, let's list out all the terms:
- [tex]\( -7 \)[/tex] (constant term)
- [tex]\( 4x^2 \)[/tex] (x squared term)
- [tex]\( 3xy^2 \)[/tex] (xy squared term)
- [tex]\( 6x^3 \)[/tex] (x cubed term)
- [tex]\( -x^2 \)[/tex] (x squared term)
- [tex]\( xy^2 \)[/tex] (xy squared term)
Next, combine the terms with the same degree and variables:
1. Constant term:
- The constant term in [tex]\( P(x, y) \)[/tex] is [tex]\(-7\)[/tex].
- There is no constant term in [tex]\( Q(x, y) \)[/tex].
[tex]\[ -7 \][/tex]
2. [tex]\( x^2 \)[/tex] term:
- From [tex]\( P(x, y) \)[/tex]: [tex]\( 4x^2 \)[/tex]
- From [tex]\( Q(x, y) \)[/tex]: [tex]\( -x^2 \)[/tex]
- Combine them: [tex]\( 4x^2 - x^2 = 3x^2 \)[/tex]
[tex]\[ 3x^2 \][/tex]
3. [tex]\( xy^2 \)[/tex] term:
- From [tex]\( P(x, y) \)[/tex]: [tex]\( 3xy^2 \)[/tex]
- From [tex]\( Q(x, y) \)[/tex]: [tex]\( xy^2 \)[/tex]
- Combine them: [tex]\( 3xy^2 + xy^2 = 4xy^2 \)[/tex]
[tex]\[ 4xy^2 \][/tex]
4. [tex]\( x^3 \)[/tex] term:
- From [tex]\( Q(x, y) \)[/tex]: [tex]\( 6x^3 \)[/tex]
- There is no [tex]\( x^3 \)[/tex] term in [tex]\( P(x, y) \)[/tex].
[tex]\[ 6x^3 \][/tex]
Now, putting all these combined terms together, we have:
[tex]\[ P(x, y) + Q(x, y) = 6x^3 + 3x^2 + 4xy^2 - 7 \][/tex]
Thus, the sum of the polynomials [tex]\( \left(-7 + 4x^2 + 3xy^2\right) \)[/tex] and [tex]\( \left(6x^3 - x^2 + xy^2\right) \)[/tex] is:
[tex]\[ 6x^3 + 3x^2 + 4xy^2 - 7 \][/tex]
