To add the given polynomials:
[tex]\[ \left(-4y^3 - 5y + 16\right) + \left(4y^2 - y + 9\right) \][/tex]
1. Identify the terms and their coefficients:
- The first polynomial [tex]\( -4y^3 - 5y + 16 \)[/tex] has the terms:
- [tex]\( -4y^3 \)[/tex] (coefficient -4)
- [tex]\( -5y \)[/tex] (coefficient -5)
- [tex]\( 16 \)[/tex] (constant term)
- The second polynomial [tex]\( 4y^2 - y + 9 \)[/tex] has the terms:
- [tex]\( 4y^2 \)[/tex] (coefficient 4)
- [tex]\( -y \)[/tex] (coefficient -1)
- [tex]\( 9 \)[/tex] (constant term)
2. Align the polynomials by like terms:
[tex]\[
\begin{aligned}
& -4y^3 + 0y^2 - 5y + 16 \\
& +\quad 0y^3 + 4y^2 - 1y + 9 \\
\end{aligned}
\][/tex]
3. Add the coefficients of the like terms:
- For [tex]\( y^3 \)[/tex] term: [tex]\( -4 + 0 = -4 \)[/tex]
- For [tex]\( y^2 \)[/tex] term: [tex]\( 0 + 4 = 4 \)[/tex]
- For [tex]\( y \)[/tex] term: [tex]\( -5 - 1 = -6 \)[/tex]
- For the constant term: [tex]\( 16 + 9 = 25 \)[/tex]
4. Combine the results into a single polynomial:
[tex]\[
-4y^3 + 4y^2 - 6y + 25
\][/tex]
Thus, the expanded polynomial in standard form is:
[tex]\[
\left(-4y^3 - 5y + 16\right) + \left(4y^2 - y + 9\right) = -4y^3 + 4y^2 - 6y + 25
\][/tex]
