Solve 27x^3-108x^2+144x-64=0

$$27 { x }^{ 3 } -108 { x }^{ 2 } +144x-64=0$$

Solve for x

$$x = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333$$

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 -64 and q divides the leading coefficient 27. List all candidates \frac{p}{q}.

$$±\frac{64}{27},±\frac{64}{9},±\frac{64}{3},±64,±\frac{32}{27},±\frac{32}{9},±\frac{32}{3},±32,±\frac{16}{27},±\frac{16}{9},±\frac{16}{3},±16,±\frac{8}{27},±\frac{8}{9},±\frac{8}{3},±8,±\frac{4}{27},±\frac{4}{9},±\frac{4}{3},±4,±\frac{2}{27},±\frac{2}{9},±\frac{2}{3},±2,±\frac{1}{27},±\frac{1}{9},±\frac{1}{3},±1$$

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.

$$x=\frac{4}{3}$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 27x^{3}-108x^{2}+144x-64 by 3\left(x-\frac{4}{3}\right)=3x-4 to get 9x^{2}-24x+16. Solve the equation where the result equals to 0.

$$9x^{2}-24x+16=0$$

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 9 for a, -24 for b, and 16 for c in the quadratic formula.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 9\times 16}}{2\times 9}$$

Do the calculations.

$$x=\frac{24±0}{18}$$

Solutions are the same.

$$x=\frac{4}{3}$$