Certainly! Here is the step-by-step process to add the given polynomials and represent the answer in standard form.
Given polynomials:
[tex]\[ \left(-4 y^3 - 5 y + 16\right) + \left(4 y^2 - y + 9\right) \][/tex]
To add these polynomials, we need to combine like terms. Like terms are terms that have the same variable raised to the same power.
### Step-by-Step Solution:
1. Identify and Arrange Like Terms:
- [tex]\( -4 y^3 \)[/tex] (from the first polynomial)
- [tex]\( 4 y^2 \)[/tex] (from the second polynomial)
- [tex]\( -5 y \)[/tex] (from the first polynomial)
- [tex]\( -y \)[/tex] (from the second polynomial)
- [tex]\( 16 \)[/tex] (from the first polynomial)
- [tex]\( 9 \)[/tex] (from the second polynomial)
2. Combine Like Terms:
- For [tex]\( y^3 \)[/tex] terms: There is only one term, [tex]\( -4 y^3 \)[/tex].
- For [tex]\( y^2 \)[/tex] terms: There is only one term, [tex]\( 4 y^2 \)[/tex].
- For [tex]\( y \)[/tex] terms:
[tex]\[ -5 y - y = -6 y \][/tex]
- For constant terms:
[tex]\[ 16 + 9 = 25 \][/tex]
3. Write the Combined Result:
- Combining all the like terms, we have:
[tex]\[ -4 y^3 + 4 y^2 - 6 y + 25 \][/tex]
### Final Answer in Expanded Form:
[tex]\[ \left(-4 y^3 - 5 y + 16\right) + \left(4 y^2 - y + 9\right) = -4 y^3 + 4 y^2 - 6 y + 25 \][/tex]
So, the expanded polynomial in standard form is:
[tex]\[ \boxed{-4y^3 + 4y^2 - 6y + 25} \][/tex]
