Solve 36x^3+144x^2-x-4=0

$$36 x ^ { 3 } + 144 x ^ { 2 } - x - 4 = 0$$

Solve for x

$$x=-\frac{1}{6}\approx -0.166666667$$

$$x=\frac{1}{6}\approx 0.166666667$$

$$x=-4$$

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 -4 and q divides the leading coefficient 36. List all candidates \frac{p}{q}.

$$±\frac{1}{9},±\frac{2}{9},±\frac{1}{3},±\frac{4}{9},±\frac{2}{3},±1,±\frac{4}{3},±2,±4,±\frac{1}{18},±\frac{1}{6},±\frac{1}{2},±\frac{1}{36},±\frac{1}{12},±\frac{1}{4}$$

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.

$$x=-4$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 36x^{3}+144x^{2}-x-4 by x+4 to get 36x^{2}-1. Solve the equation where the result equals to 0.

$$36x^{2}-1=0$$

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 36 for a, 0 for b, and -1 for c in the quadratic formula.

$$x=\frac{0±\sqrt{0^{2}-4\times 36\left(-1\right)}}{2\times 36}$$

Do the calculations.

$$x=\frac{0±12}{72}$$

Solve the equation 36x^{2}-1=0 when ± is plus and when ± is minus.

$$x=-\frac{1}{6} x=\frac{1}{6}$$

List all found solutions.

$$x=-4 x=-\frac{1}{6} x=\frac{1}{6}$$