Solve =x^4-162x^2+6561

$$= x ^ { 4 } - 162 x ^ { 2 } + 6561$$

Factor

$$\left(x-9\right)^{2}\left(x+9\right)^{2}$$

Solution Steps

To factor the expression, solve the equation where it equals to 0.

$$x^{4}-162x^{2}+6561=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 6561 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±6561,±2187,±729,±243,±81,±27,±9,±3,±1$$

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.

$$x=9$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{4}-162x^{2}+6561 by x-9 to get x^{3}+9x^{2}-81x-729. To factor the result, solve the equation where it equals to 0.

$$x^{3}+9x^{2}-81x-729=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -729 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±729,±243,±81,±27,±9,±3,±1$$

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.

$$x=9$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}+9x^{2}-81x-729 by x-9 to get x^{2}+18x+81. To factor the result, solve the equation where it equals to 0.

$$x^{2}+18x+81=0$$

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, 18 for b, and 81 for c in the quadratic formula.

$$x=\frac{-18±\sqrt{18^{2}-4\times 1\times 81}}{2}$$

Do the calculations.

$$x=\frac{-18±0}{2}$$

Solutions are the same.

$$x=-9$$

Rewrite the factored expression using the obtained roots.

$$\left(x-9\right)^{2}\left(x+9\right)^{2}$$

Evaluate

$$\left(x^{2}-81\right)^{2}$$