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).
- 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).
