The given equation is:

\[
v^6 - 25v^4 = 0
\]

We can factor this equation to solve for \( v \):

### Step 1: Factor out \( v^4 \) from both terms:
\[
v^4(v^2 - 25) = 0
\]

### Step 2: Solve each factor separately.

1. \( v^4 = 0 \)

   Taking the fourth root of both sides:
   \[
   v = 0
   \]

2. \( v^2 - 25 = 0 \)

   Rearranging the equation:
   \[
   v^2 = 25
   \]

   Taking the square root of both sides:
   \[
   v = \pm 5
   \]

### Final Solutions:
The solutions to the equation \( v^6 - 25v^4 = 0 \) are:
\[
v = 0, \, v = 5, \, v = -5
\]

Would you like more details or clarifications on this?

Here are 5 related questions to expand on this topic:
1. How do you solve higher-degree polynomials by factoring?
2. What are the steps to factor a quadratic expression?
3. How do you solve equations involving even powers like \( v^4 \) and \( v^6 \)?
4. What is the difference between real and complex solutions in polynomial equations?
5. How do different roots (e.g., square roots vs. fourth roots) affect the number of solutions?

**Tip:** When factoring, always look for common terms to simplify the expression!

Learn how to solve the polynomial equation v^6 - 25v^4 = 0 by factoring. This detailed step-by-step solution covers key algebraic concepts such as factoring polynomials and applying the zero product property. Suitable for high school students in Grades 10-12.

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