Solve lambda^3-27lambda^2+243lambda-729=0

$$\lambda ^ { 3 } - 27 \lambda ^ { 2 } + 243 \lambda - 729 = 0$$

Solve for λ

$$\lambda =9$$

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 -729 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±729,±243,±81,±27,±9,±3,±1$$

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.

$$\lambda =9$$

By Factor theorem, \lambda -k is a factor of the polynomial for each root k. Divide \lambda ^{3}-27\lambda ^{2}+243\lambda -729 by \lambda -9 to get \lambda ^{2}-18\lambda +81. Solve the equation where the result equals to 0.

$$\lambda ^{2}-18\lambda +81=0$$

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, -18 for b, and 81 for c in the quadratic formula.

$$\lambda =\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 1\times 81}}{2}$$

Do the calculations.

$$\lambda =\frac{18±0}{2}$$

Solutions are the same.

$$\lambda =9$$