Solve t^3+t-2=0

$$t ^ { 3 } + t - 2 = 0$$

Solve for t

$$t=1$$

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 -2 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±2,±1$$

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.

$$t=1$$

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

$$t^{2}+t+2=0$$

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, 1 for b, and 2 for c in the quadratic formula.

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

Do the calculations.

$$t=\frac{-1±\sqrt{-7}}{2}$$

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

$$t\in \emptyset$$

List all found solutions.

$$t=1$$