Solve z^3+3z^2+4z-8=0

$$z ^ { 3 } + 3 z ^ { 2 } + 4 z - 8 = 0$$

Solve for z (complex solution)

$$z=-2-2i$$

$$z=1$$

$$z=-2+2i$$

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 -8 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±8,±4,±2,±1$$

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.

$$z=1$$

By Factor theorem, z-k is a factor of the polynomial for each root k. Divide z^{3}+3z^{2}+4z-8 by z-1 to get z^{2}+4z+8. Solve the equation where the result equals to 0.

$$z^{2}+4z+8=0$$

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, 4 for b, and 8 for c in the quadratic formula.

$$z=\frac{-4±\sqrt{4^{2}-4\times 1\times 8}}{2}$$

Do the calculations.

$$z=\frac{-4±\sqrt{-16}}{2}$$

Solve the equation z^{2}+4z+8=0 when ± is plus and when ± is minus.

$$z=-2-2i z=-2+2i$$

List all found solutions.

$$z=1 z=-2-2i z=-2+2i$$

Solve for z

$$z=1$$

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 -8 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±8,±4,±2,±1$$

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.

$$z=1$$

By Factor theorem, z-k is a factor of the polynomial for each root k. Divide z^{3}+3z^{2}+4z-8 by z-1 to get z^{2}+4z+8. Solve the equation where the result equals to 0.

$$z^{2}+4z+8=0$$

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, 4 for b, and 8 for c in the quadratic formula.

$$z=\frac{-4±\sqrt{4^{2}-4\times 1\times 8}}{2}$$

Do the calculations.

$$z=\frac{-4±\sqrt{-16}}{2}$$

Since the square root of a negative number is not defined in the real field, there are no solutions.

$$z\in \emptyset$$

List all found solutions.

$$z=1$$