Answer :
- Find the greatest common factor (GCF) of the coefficients: GCF(60, 50) = 10.
- Find the GCF of the variable terms: GCF($x^3$, $x^5$) = $x^3$.
- Factor out the GCF $10x^3$ from the expression: $60x^3 + 50x^5 = 10x^3(6 + 5x^2)$.
- The completely factored expression is $\boxed{10x^3(6 + 5x^2)}$.
### Explanation
1. Understanding the Problem
We are asked to factor the expression $60x^3 + 50x^5$ completely. This means we want to find the greatest common factor (GCF) of the terms and factor it out.
2. Finding the GCF of the Coefficients
First, let's find the GCF of the coefficients, 60 and 50. The GCF of 60 and 50 is 10.
3. Finding the GCF of the Variable Terms
Next, let's find the GCF of the variable terms, $x^3$ and $x^5$. The GCF of $x^3$ and $x^5$ is $x^3$ because it is the lowest power of $x$ that divides both terms.
4. Factoring out the GCF
Now, we can factor out the GCF, which is $10x^3$, from the expression: $$60x^3 + 50x^5 = 10x^3(6) + 10x^3(5x^2) = 10x^3(6 + 5x^2)$$.
5. Final Answer
Thus, the completely factored expression is $10x^3(6 + 5x^2)$.
### Examples
Factoring expressions is a fundamental skill in algebra and is used in many real-world applications. For example, if you are designing a rectangular garden and want to express the area in terms of its length and width, you might end up with an expression that needs to be factored to find the dimensions. Similarly, in physics, factoring can help simplify equations related to motion or energy. Factoring allows us to break down complex expressions into simpler, more manageable parts, making it easier to solve problems and understand relationships between variables.
- Find the GCF of the variable terms: GCF($x^3$, $x^5$) = $x^3$.
- Factor out the GCF $10x^3$ from the expression: $60x^3 + 50x^5 = 10x^3(6 + 5x^2)$.
- The completely factored expression is $\boxed{10x^3(6 + 5x^2)}$.
### Explanation
1. Understanding the Problem
We are asked to factor the expression $60x^3 + 50x^5$ completely. This means we want to find the greatest common factor (GCF) of the terms and factor it out.
2. Finding the GCF of the Coefficients
First, let's find the GCF of the coefficients, 60 and 50. The GCF of 60 and 50 is 10.
3. Finding the GCF of the Variable Terms
Next, let's find the GCF of the variable terms, $x^3$ and $x^5$. The GCF of $x^3$ and $x^5$ is $x^3$ because it is the lowest power of $x$ that divides both terms.
4. Factoring out the GCF
Now, we can factor out the GCF, which is $10x^3$, from the expression: $$60x^3 + 50x^5 = 10x^3(6) + 10x^3(5x^2) = 10x^3(6 + 5x^2)$$.
5. Final Answer
Thus, the completely factored expression is $10x^3(6 + 5x^2)$.
### Examples
Factoring expressions is a fundamental skill in algebra and is used in many real-world applications. For example, if you are designing a rectangular garden and want to express the area in terms of its length and width, you might end up with an expression that needs to be factored to find the dimensions. Similarly, in physics, factoring can help simplify equations related to motion or energy. Factoring allows us to break down complex expressions into simpler, more manageable parts, making it easier to solve problems and understand relationships between variables.
