Solve 12x^4-4x^3-41x^2-4x+12=0

$$12 { x }^{ 4 } -4 { x }^{ 3 } -41 { x }^{ 2 } -4x+12=0$$

Solve for x

$$x=\frac{1}{2}=0.5$$

$$x=2$$

$$x=-\frac{2}{3}\approx -0.666666667$$

$$x = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

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 12 and q divides the leading coefficient 12. List all candidates \frac{p}{q}.

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

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.

$$x=2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 12x^{4}-4x^{3}-41x^{2}-4x+12 by x-2 to get 12x^{3}+20x^{2}-x-6. Solve the equation where the result equals to 0.

$$12x^{3}+20x^{2}-x-6=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -6 and q divides the leading coefficient 12. List all candidates \frac{p}{q}.

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

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.

$$x=\frac{1}{2}$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 12x^{3}+20x^{2}-x-6 by 2\left(x-\frac{1}{2}\right)=2x-1 to get 6x^{2}+13x+6. Solve the equation where the result equals to 0.

$$6x^{2}+13x+6=0$$

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 6 for a, 13 for b, and 6 for c in the quadratic formula.

$$x=\frac{-13±\sqrt{13^{2}-4\times 6\times 6}}{2\times 6}$$

Do the calculations.

$$x=\frac{-13±5}{12}$$

Solve the equation 6x^{2}+13x+6=0 when ± is plus and when ± is minus.

$$x=-\frac{3}{2} x=-\frac{2}{3}$$

List all found solutions.

$$x=2 x=\frac{1}{2} x=-\frac{3}{2} x=-\frac{2}{3}$$