Answer :
- Find the greatest common factor (GCF) of the coefficients: GCF(15, 25) = 5.
- Find the greatest common factor of the variable terms: GCF($x^3$, $x^2$) = $x^2$.
- Factor out the GCF $5x^2$ from the expression: $15x^3 + 25x^2 = 5x^2(3x + 5)$.
- The factored expression is $\boxed{5x^2(3x + 5)}$.
### Explanation
1. Understanding the Problem
We are asked to factor the expression $15x^3 + 25x^2$. Factoring involves finding the greatest common factor (GCF) of the terms and then rewriting the expression as a product of the GCF and the remaining terms.
2. Finding GCF of Coefficients
First, let's find the GCF of the coefficients, 15 and 25. The factors of 15 are 1, 3, 5, and 15. The factors of 25 are 1, 5, and 25. The greatest common factor of 15 and 25 is 5.
3. Finding GCF of Variables
Next, let's find the GCF of the variable terms, $x^3$ and $x^2$. The GCF of $x^3$ and $x^2$ is $x^2$ because $x^2$ is the highest power of $x$ that divides both terms.
4. Factoring out the GCF
Now, we can factor out the GCF, which is $5x^2$, from the original expression: $$15x^3 + 25x^2 = 5x^2(3x) + 5x^2(5)$$. We can rewrite this as: $$15x^3 + 25x^2 = 5x^2(3x + 5)$$.
5. Final Answer
Therefore, the factored form of $15x^3 + 25x^2$ is $5x^2(3x + 5)$.
### Examples
Factoring polynomials is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to simplify complex equations when designing structures or analyzing systems. In finance, factoring can be used to model and predict investment growth. Understanding how to factor expressions like this helps in simplifying problems and finding solutions more efficiently.
- Find the greatest common factor of the variable terms: GCF($x^3$, $x^2$) = $x^2$.
- Factor out the GCF $5x^2$ from the expression: $15x^3 + 25x^2 = 5x^2(3x + 5)$.
- The factored expression is $\boxed{5x^2(3x + 5)}$.
### Explanation
1. Understanding the Problem
We are asked to factor the expression $15x^3 + 25x^2$. Factoring involves finding the greatest common factor (GCF) of the terms and then rewriting the expression as a product of the GCF and the remaining terms.
2. Finding GCF of Coefficients
First, let's find the GCF of the coefficients, 15 and 25. The factors of 15 are 1, 3, 5, and 15. The factors of 25 are 1, 5, and 25. The greatest common factor of 15 and 25 is 5.
3. Finding GCF of Variables
Next, let's find the GCF of the variable terms, $x^3$ and $x^2$. The GCF of $x^3$ and $x^2$ is $x^2$ because $x^2$ is the highest power of $x$ that divides both terms.
4. Factoring out the GCF
Now, we can factor out the GCF, which is $5x^2$, from the original expression: $$15x^3 + 25x^2 = 5x^2(3x) + 5x^2(5)$$. We can rewrite this as: $$15x^3 + 25x^2 = 5x^2(3x + 5)$$.
5. Final Answer
Therefore, the factored form of $15x^3 + 25x^2$ is $5x^2(3x + 5)$.
### Examples
Factoring polynomials is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to simplify complex equations when designing structures or analyzing systems. In finance, factoring can be used to model and predict investment growth. Understanding how to factor expressions like this helps in simplifying problems and finding solutions more efficiently.
