Solve x^4+4x^3-22x^2-4x+21=0

$$x ^ { 4 } + 4 x ^ { 3 } - 22 x ^ { 2 } - 4 x + 21 = 0$$

Solve for x

$$x=1$$

$$x=3$$

$$x=-1$$

$$x=-7$$

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

$$±21,±7,±3,±1$$

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.

$$x=1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{4}+4x^{3}-22x^{2}-4x+21 by x-1 to get x^{3}+5x^{2}-17x-21. Solve the equation where the result equals to 0.

$$x^{3}+5x^{2}-17x-21=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -21 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±21,±7,±3,±1$$

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.

$$x=-1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}+5x^{2}-17x-21 by x+1 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.

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

Do the calculations.

$$x=\frac{-4±10}{2}$$

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

$$x=-7 x=3$$

List all found solutions.

$$x=1 x=-1 x=-7 x=3$$