Solve z^3+3z^2-25z-75=0

$$z ^ { 3 } + 3 z ^ { 2 } - 25 z - 75 = 0$$

Solve for z

$$z=-3$$

$$z=5$$

$$z=-5$$

Solution Steps

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -75 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±75,±25,±15,±5,±3,±1$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$z=-3$$

By Factor theorem, z-k is a factor of the polynomial for each root k. Divide z^{3}+3z^{2}-25z-75 by z+3 to get z^{2}-25. Solve the equation where the result equals to 0.

$$z^{2}-25=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, 0 for b, and -25 for c in the quadratic formula.

$$z=\frac{0±\sqrt{0^{2}-4\times 1\left(-25\right)}}{2}$$

Do the calculations.

$$z=\frac{0±10}{2}$$

Solve the equation z^{2}-25=0 when ± is plus and when ± is minus.

$$z=-5 z=5$$

List all found solutions.

$$z=-3 z=-5 z=5$$