High School

Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ Simplify the expression:

[tex]4x^5 - 25x^3[/tex]

Answer :

- Factor out the greatest common factor: $x^3(4x^2 - 25)$.
- Recognize the difference of squares: $4x^2 - 25 = (2x - 5)(2x + 5)$.
- Write the completely factored expression: $\boxed{x^3(2x - 5)(2x + 5)}$.

### Explanation
1. Understanding the Problem
We are given the expression $4x^5 - 25x^3$ and asked to factor it completely.

2. Factoring out the GCF
First, we identify the greatest common factor (GCF) of the terms. Both terms have $x$ as a factor. The lowest power of $x$ in the expression is $x^3$. So, we factor out $x^3$ from the expression: $$4x^5 - 25x^3 = x^3(4x^2 - 25)$$

3. Factoring the Difference of Squares
Now, we look at the expression inside the parentheses: $4x^2 - 25$. This is a difference of squares, which can be factored as $(a^2 - b^2) = (a - b)(a + b)$. In our case, $a^2 = 4x^2$ and $b^2 = 25$. Thus, $a = 2x$ and $b = 5$. So, we can factor $4x^2 - 25$ as:$$4x^2 - 25 = (2x - 5)(2x + 5)$$

4. Final Factored Form
Finally, we substitute this back into our expression:$$4x^5 - 25x^3 = x^3(2x - 5)(2x + 5)$$This is the completely factored form of the given expression.

### Examples
Factoring polynomials is a fundamental skill in algebra. It's used in many areas, such as solving equations, simplifying expressions, and analyzing functions. For example, if you have a function representing the profit of a business, factoring it can help you find the break-even points (where the profit is zero).