Solve (x^4-40x^2+144)

$$( x ^ { 4 } - 40 x ^ { 2 } + 144 )$$

Factor

$$\left(x-6\right)\left(x-2\right)\left(x+2\right)\left(x+6\right)$$

Solution Steps

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

$$x^{4}-40x^{2}+144=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 144 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±144,±72,±48,±36,±24,±18,±16,±12,±9,±8,±6,±4,±3,±2,±1$$

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}-40x^{2}+144 by x-2 to get x^{3}+2x^{2}-36x-72. To factor the result, solve the equation where it equals to 0.

$$x^{3}+2x^{2}-36x-72=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -72 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±72,±36,±24,±18,±12,±9,±8,±6,±4,±3,±2,±1$$

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^{3}+2x^{2}-36x-72 by x+2 to get x^{2}-36. To factor the result, solve the equation where it equals to 0.

$$x^{2}-36=0$$

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, 0 for b, and -36 for c in the quadratic formula.

$$x=\frac{0±\sqrt{0^{2}-4\times 1\left(-36\right)}}{2}$$

Do the calculations.

$$x=\frac{0±12}{2}$$

Solve the equation x^{2}-36=0 when ± is plus and when ± is minus.

$$x=-6 x=6$$

Rewrite the factored expression using the obtained roots.

$$\left(x-6\right)\left(x-2\right)\left(x+2\right)\left(x+6\right)$$

Evaluate

$$x^{4}-40x^{2}+144$$