Solve the Polynomial Equation: x^8 - 25x^6 = 0

To solve the equation $x^8 - 25x^6 = 0$, we start by noticing that both terms share a common factor of $x^6$. We can factor out $x^6$ from the expression:

$x^6(x^2 - 25) = 0$

According to the zero-product property, a product is zero if and only if at least one of the factors is zero. Therefore, we have two separate equations to solve:

• $x^6 = 0$
• $x^2 - 25 = 0$

For $x^6 = 0$:

$x = 0$

For $x^2 - 25 = 0$, this can be seen as a difference of squares, which factors as:

$(x - 5)(x + 5) = 0$

Again, using the zero-product property, we solve the factors:

• $x - 5 = 0$ gives $x = 5$
• $x + 5 = 0$ gives $x = -5$

The solutions to the equation are therefore $x = 0, x = 5,$ and $x = -5$.

The correct answer choice is "Answers a + b", where $\pm 5$ and $0$ are included as solutions.