We are given the result of adding two polynomials is:
[tex]8d^5-3c^3d^2+5c^2d^3-4cd^4+9[/tex]
One of the polynomials that is added is:
[tex]2d^5-c^3d^2+8cd^4+1[/tex]
To determine the polynomial that was added we will use a polynomial with the same variables that appear on each polynomial but with unknown coefficients, like this:
[tex]2d^5-c^3d^2+8cd^4+1+(xd^5+yc^3d^2+wc^2d^3+zcd^4+u)[/tex]
Now we associate like terms:
[tex](2+x)d^5+(-1+y)c^3d^2+wc^2d^3+(8+z)cd^4+(1+u)[/tex]
Now, each of the new coefficients must be equal to the coefficients in the resulting polynomial, therefore, we have for the first term:
[tex]\begin{gathered} 2+x=8 \\ x=8-2 \\ x=6 \end{gathered}[/tex]
For the second term:
[tex]\begin{gathered} -1+y=-3 \\ y=-3+1 \\ y=-2 \end{gathered}[/tex]
For the third term:
[tex]\begin{gathered} w=5 \\ \end{gathered}[/tex]
For the fourth term:
[tex]\begin{gathered} 8+z=-4 \\ z=-4-8 \\ z=-12 \end{gathered}[/tex]
For the last term:
[tex]\begin{gathered} 1+u=9 \\ u=9-1 \\ u=8 \end{gathered}[/tex]
Now that we have the coefficients we can substitute in the polynomial we added, we get:
[tex]6d^5-2c^3d^2+5c^2d^3-12cd^4+8[/tex]
