Question

What is the sum of the roots of a polynomial shown below


f(x)=3x^3+12x^2+3x-18

Answer 1

Let a, b, and c be the roots of this polynomial. Then the sum of the roots would be a+b+c. If we were to factor the polynomial, we would get \[f(x)=3x^3+12x^2+3x-18=3(x-a)(x-b)(x-c)\] Note however that if you were to expand the right hand side, you would get \[3(x-a)(x-b)(x-c)=3(x^3-(a+b+c)x^2+ \cdots)=3x^3-3(a+b+c)x^2 + \cdots\] Thus, you have both \[f(x)=3x^3+12x^2+\cdots=3x^3-3(a+b+c)x^2+\cdots\] Thus, by equating the coefficients of the x^2 term, you have \[12=-3(a+b+c) \implies a+b+c=-4\] Thus, the sum of the roots is -4 (even though you don't even know what the roots are!!!).