Solve 5x^3-11x^2+7x-1=0

$$5 x ^ { 3 } - 11 x ^ { 2 } + 7 x - 1 = 0$$

Solve for x

$$x=1$$

$$x=\frac{1}{5}=0.2$$

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 5. List all candidates \frac{p}{q}.

$$±\frac{1}{5},±1$$

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.

$$x=1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 5x^{3}-11x^{2}+7x-1 by x-1 to get 5x^{2}-6x+1. Solve the equation where the result equals to 0.

$$5x^{2}-6x+1=0$$

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 5 for a, -6 for b, and 1 for c in the quadratic formula.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 5\times 1}}{2\times 5}$$

Do the calculations.

$$x=\frac{6±4}{10}$$

Solve the equation 5x^{2}-6x+1=0 when ± is plus and when ± is minus.

$$x=\frac{1}{5} x=1$$

List all found solutions.

$$x=1 x=\frac{1}{5}$$