Solve lambda^3-lambda^2-lambda+1=0

$$\lambda ^ { 3 } - \lambda ^ { 2 } - \lambda + 1 = 0$$

Solve for λ

$$\lambda =1$$

$$\lambda =-1$$

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

$$±1$$

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.

$$\lambda =1$$

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

$$\lambda ^{2}-1=0$$

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 -1 for c in the quadratic formula.

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

Do the calculations.

$$\lambda =\frac{0±2}{2}$$

Solve the equation \lambda ^{2}-1=0 when ± is plus and when ± is minus.

$$\lambda =-1 \lambda =1$$

List all found solutions.

$$\lambda =1 \lambda =-1$$