To find the other addend polynomial, we need to subtract one of the given addends from the sum of the polynomials. Let's denote the sum of the polynomials as [tex]\( S \)[/tex] and the known addend as [tex]\( A_1 \)[/tex]. We need to find the unknown addend [tex]\( A_2 \)[/tex].
Given:
[tex]\[ S = 8d^5 - 3c^3d^2 + 5c^2d^3 - 4cd^4 + 9 \][/tex]
[tex]\[ A_1 = 2d^5 - c^3d^2 + 8cd^4 + 1 \][/tex]
We need to calculate each coefficient of [tex]\( A_2 \)[/tex] where:
[tex]\[ A_2 = S - A_1 \][/tex]
This involves subtracting the corresponding coefficients of [tex]\( A_1 \)[/tex] from [tex]\( S \)[/tex]:
1. Coefficient of [tex]\( d^5 \)[/tex]
[tex]\[ 8d^5 - 2d^5 = (8 - 2)d^5 = 6d^5 \][/tex]
2. Coefficient of [tex]\( c^3d^2 \)[/tex]
[tex]\[ -3c^3d^2 - (-1c^3d^2) = (-3 + 1)c^3d^2 = -2c^3d^2 \][/tex]
3. Coefficient of [tex]\( c^2d^3 \)[/tex]
[tex]\[ 5c^2d^3 - 0c^2d^3 = 5c^2d^3 \][/tex]
4. Coefficient of [tex]\( cd^4 \)[/tex]
[tex]\[ -4cd^4 - 8cd^4 = (-4 - 8)cd^4 = -12cd^4 \][/tex]
5. Constant term
[tex]\[ 9 - 1 = 8 \][/tex]
So, the other addend polynomial [tex]\( A_2 \)[/tex] is:
[tex]\[ 6d^5 - 2c^3d^2 + 5c^2d^3 - 12cd^4 + 8 \][/tex]
Thus, the solution to the problem is:
[tex]\[ 6d^5 - 2c^3d^2 + 5c^2d^3 - 12cd^4 + 8 \][/tex]
Which corresponds to the first option provided:
[tex]\[ \boxed{6d^5 - 2c^3d^2 + 5c^2d^3 - 12cd^4 + 8} \][/tex]
