Solve -x^6+6x^4-9x^2+4

$$- { x }^{ 6 } +6 { x }^{ 4 } -9 { x }^{ 2 } +4$$

Factor

$$\left(-x-1\right)\left(x-2\right)\left(x+1\right)\left(x+2\right)\left(x-1\right)^{2}$$

Solution Steps

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

$$-x^{6}+6x^{4}-9x^{2}+4=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 4 and q divides the leading coefficient -1. List all candidates \frac{p}{q}.

$$±4,±2,±1$$

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.

$$x=1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide -x^{6}+6x^{4}-9x^{2}+4 by x-1 to get -x^{5}-x^{4}+5x^{3}+5x^{2}-4x-4. To factor the result, solve the equation where it equals to 0.

$$-x^{5}-x^{4}+5x^{3}+5x^{2}-4x-4=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -4 and q divides the leading coefficient -1. List all candidates \frac{p}{q}.

$$±4,±2,±1$$

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.

$$x=-1$$

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

$$-x^{4}+5x^{2}-4=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -4 and q divides the leading coefficient -1. List all candidates \frac{p}{q}.

$$±4,±2,±1$$

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.

$$x=-1$$

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

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

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -4 and q divides the leading coefficient -1. List all candidates \frac{p}{q}.

$$±4,±2,±1$$

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}+x^{2}+4x-4 by x-2 to get -x^{2}-x+2. To factor the result, solve the equation where it equals to 0.

$$-x^{2}-x+2=0$$

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, -1 for b, and 2 for c in the quadratic formula.

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

Do the calculations.

$$x=\frac{1±3}{-2}$$

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

$$x=1 x=-2$$

Rewrite the factored expression using the obtained roots.

$$\left(x-2\right)\left(x-1\right)\left(x+2\right)\left(-x+1\right)\left(x+1\right)^{2}$$

Evaluate

$$4-9x^{2}+6x^{4}-x^{6}$$