Solve 7x^3-3x^2-3x-1=0

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

Solve for x (complex solution)

$$x=\frac{-\sqrt{3}i-2}{7}\approx -0.285714286-0.24743583i$$

$$x=1$$

$$x=\frac{-2+\sqrt{3}i}{7}\approx -0.285714286+0.24743583i$$

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

$$±\frac{1}{7},±1$$

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 7x^{3}-3x^{2}-3x-1 by x-1 to get 7x^{2}+4x+1. Solve the equation where the result equals to 0.

$$7x^{2}+4x+1=0$$

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 7 for a, 4 for b, and 1 for c in the quadratic formula.

$$x=\frac{-4±\sqrt{4^{2}-4\times 7\times 1}}{2\times 7}$$

Do the calculations.

$$x=\frac{-4±\sqrt{-12}}{14}$$

Solve the equation 7x^{2}+4x+1=0 when ± is plus and when ± is minus.

$$x=\frac{-\sqrt{3}i-2}{7} x=\frac{-2+\sqrt{3}i}{7}$$

List all found solutions.

$$x=1 x=\frac{-\sqrt{3}i-2}{7} x=\frac{-2+\sqrt{3}i}{7}$$

Solve for x

$$x=1$$

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

$$±\frac{1}{7},±1$$

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 7x^{3}-3x^{2}-3x-1 by x-1 to get 7x^{2}+4x+1. Solve the equation where the result equals to 0.

$$7x^{2}+4x+1=0$$

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 7 for a, 4 for b, and 1 for c in the quadratic formula.

$$x=\frac{-4±\sqrt{4^{2}-4\times 7\times 1}}{2\times 7}$$

Do the calculations.

$$x=\frac{-4±\sqrt{-12}}{14}$$

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

$$x\in \emptyset$$

List all found solutions.

$$x=1$$