Solve z^3-2z^2/z^2+7z-18=-12z-27/z+9

$$\frac { z ^ { 3 } - 2 z ^ { 2 } } { z ^ { 2 } + 7 z - 18 } = \frac { - 12 z - 27 } { z + 9 }$$

Solve for z

$$z=-3$$

Solution Steps

Variable z cannot be equal to any of the values -9,2 since division by zero is not defined. Multiply both sides of the equation by \left(z-2\right)\left(z+9\right), the least common multiple of z^{2}+7z-18,z+9.

$$z^{3}-2z^{2}=\left(z-2\right)\left(-12z-27\right)$$

Use the distributive property to multiply z-2 by -12z-27 and combine like terms.

$$z^{3}-2z^{2}=-12z^{2}-3z+54$$

Add 12z^{2} to both sides.

$$z^{3}-2z^{2}+12z^{2}=-3z+54$$

Combine -2z^{2} and 12z^{2} to get 10z^{2}.

$$z^{3}+10z^{2}=-3z+54$$

Add 3z to both sides.

$$z^{3}+10z^{2}+3z=54$$

Subtract 54 from both sides.

$$z^{3}+10z^{2}+3z-54=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -54 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±54,±27,±18,±9,±6,±3,±2,±1$$

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.

$$z=2$$

By Factor theorem, z-k is a factor of the polynomial for each root k. Divide z^{3}+10z^{2}+3z-54 by z-2 to get z^{2}+12z+27. Solve the equation where the result equals to 0.

$$z^{2}+12z+27=0$$

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, 12 for b, and 27 for c in the quadratic formula.

$$z=\frac{-12±\sqrt{12^{2}-4\times 1\times 27}}{2}$$

Do the calculations.

$$z=\frac{-12±6}{2}$$

Solve the equation z^{2}+12z+27=0 when ± is plus and when ± is minus.

$$z=-9 z=-3$$

Remove the values that the variable cannot be equal to.

$$z=-3$$

List all found solutions.

$$z=2 z=-9 z=-3$$

Variable z cannot be equal to any of the values 2,-9.

$$z=-3$$