Solve y^3-y^2-7y+3=0

$$y ^ { 3 } - y ^ { 2 } - 7 y + 3 = 0$$

Solve for y

$$y=-\sqrt{2}-1\approx -2.414213562$$

$$y=3$$

$$y=\sqrt{2}-1\approx 0.414213562$$

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

$$±3,±1$$

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.

$$y=3$$

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

$$y^{2}+2y-1=0$$

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

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

Do the calculations.

$$y=\frac{-2±2\sqrt{2}}{2}$$

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

$$y=-\sqrt{2}-1 y=\sqrt{2}-1$$

List all found solutions.

$$y=3 y=-\sqrt{2}-1 y=\sqrt{2}-1$$