Solve the Equation x⁶-4x⁴=0: Sixth-Degree Polynomial Problem

To solve this problem, we start by factoring the given equation:

The equation is $x^6 - 4x^4 = 0$. Notice that both terms contain a power of $x$. We can factor out the greatest common factor, which is $x^4$.

This yields $x^4(x^2 - 4) = 0$.

Next, we apply the zero-product property, which states that if a product of factors equals zero, at least one of the factors must be zero:

• First factor: $x^4 = 0$. This implies that $x = 0$.
• Second factor: $x^2 - 4 = 0$. We solve this quadratic equation by factoring further:

The quadratic equation $x^2 - 4 = 0$ can be factored using the difference of squares:

$(x - 2)(x + 2) = 0$.

Again applying the zero-product property, we set each factor equal to zero:

• For $x - 2 = 0$, $x = 2$.
• For $x + 2 = 0$, $x = -2$.

Thus, the complete set of solutions to the equation is $x = 0, x = 2, x = -2$.

Therefore, the solution to the problem is $x = 0, x = \pm 2$.