Solve x^4-4x^3-25x^2+16x+84=0

$$x ^ { 4 } - 4 x ^ { 3 } - 25 x ^ { 2 } + 16 x + 84 = 0$$

Solve for x

$$x=2$$

$$x=7$$

$$x=-2$$

$$x=-3$$

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

$$±84,±42,±28,±21,±14,±12,±7,±6,±4,±3,±2,±1$$

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.

$$x=2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{4}-4x^{3}-25x^{2}+16x+84 by x-2 to get x^{3}-2x^{2}-29x-42. Solve the equation where the result equals to 0.

$$x^{3}-2x^{2}-29x-42=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -42 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±42,±21,±14,±7,±6,±3,±2,±1$$

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.

$$x=-2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-2x^{2}-29x-42 by x+2 to get x^{2}-4x-21. Solve the equation where the result equals to 0.

$$x^{2}-4x-21=0$$

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, -4 for b, and -21 for c in the quadratic formula.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 1\left(-21\right)}}{2}$$

Do the calculations.

$$x=\frac{4±10}{2}$$

Solve the equation x^{2}-4x-21=0 when ± is plus and when ± is minus.

$$x=-3 x=7$$

List all found solutions.

$$x=2 x=-2 x=-3 x=7$$