Prove that a cubic equation x^3 + ax^2 + bx + c = 0 has 3 roots by finding the roots.

Question

Let x = r be a root of the cubic. Factor out x - r Using polynomial long division. Then use the quadratic equation

Answer 1

So x³ + ax² + bx + c = (x - r)[x² + (r + a)x + (r² + ar + b) + (r³ + ar² + br + c)/(x - r)]

Now, plugging this into the quadratic formula is not easy typing (I'm not even going to attempt putting that in this forum) but for the quadratic formula ax² + bx + c = 0, the "a" is 1, the b is (r + a) and the c is (r² + ar + b) + (r³ + ar² + br + c)/(x - r)

Remember that the quadratic formula is:

x = -b ± √(b² - 4ac)/2a so we just plug all those things into this equation.