solve x^3-15x^2+59x-45=0

Question

need to solve the cubic equation

Answer 1

x³-15x²+59x-45=0.

Note that the sum of the positive coefficients 1+59=60 and the sum of the negative coefficients -15-45=-60. The sum of the coefficients is therefore zero. This implies that x=1 is a zero (root), therefore one factor of this cubic is x-1. This enable us to reduce the cubic to a quadratic. Using synthetic division we get:

1 | 1 -15   59 -45

     1    1 -14 | 45

     1 -14  45 |   0 = x²-14x+45=(x-9)(x-5).

So the zeroes are 1, 5 and 9.