The equation $x^3 + 8x^2 - 4x + c = 0$ has three roots, one of which is the sum of the other two. What is $c$?
I notice that this equation is cubic, so it must have three roots, either all real or one real and two complex conjugates. I also notice that the coefficient of $x^3$ is 1, which means that the sum of the roots is equal to the opposite of the coefficient of $x^2$, which is -8. Therefore, if one root is the sum of the other two, then that root must be -8/2 = -4. This means that -4 is a root of the equation, so I can use synthetic division to find the other factor of the cubic equation. To do synthetic division, I write the coefficients of the equation in a row, and then write -4 below and to the left of the first coefficient. I bring down the first coefficient, which is 1, and multiply it by -4, and write the result below the second coefficient, which is 8. I get -4, and I add it to 8, and write the result below, which is 4. I repeat this process for the next coefficients, multiplying by -4 and adding to the next coefficient. I get -16, and I add it to -4, and write the result below, which is -20. I get 80, and I add it to c, and write the result below, which is c + 80. The last result is the remainder of the division, and it should be zero if -4 is a root of the equation. Therefore, I conclude that c + 80 = 0, or c = -80. 