Answer :
- Find the greatest common factor (GCF) of the coefficients: GCF(8, -48) = 8.
- Find the GCF of the variable terms: GCF($x^5$, $x^3$) = $x^3$.
- Determine the overall GCF: $8x^3$.
- Factor out the GCF from the polynomial: $8x^5 - 48x^3 = 8x^3(x^2 - 6)$.
- The completely factored polynomial is $\boxed{8x^3(x^2 - 6)}$.
### Explanation
1. Understanding the Problem
We are asked to factor the polynomial $8x^5 - 48x^3$ completely using the greatest common factor (GCF). This means we need to identify the largest factor that divides both terms of the polynomial.
2. Finding the GCF of the Coefficients
First, let's find the GCF of the coefficients, which are 8 and -48. The factors of 8 are 1, 2, 4, and 8. The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. The greatest common factor of 8 and 48 is 8.
3. Finding the GCF of the Variable Terms
Next, let's find the GCF of the variable terms, which are $x^5$ and $x^3$. The GCF of $x^5$ and $x^3$ is $x^3$ because $x^3$ is the highest power of $x$ that divides both $x^5$ and $x^3$.
4. Determining the Overall GCF
Now, we combine the GCF of the coefficients and the GCF of the variable terms to find the overall GCF of the polynomial. The GCF is $8x^3$.
5. Factoring out the GCF
We factor out the GCF from the polynomial: $$8x^5 - 48x^3 = 8x^3(x^2 - 6)$$. To verify, we can distribute $8x^3$ back into the parentheses: $8x^3 * x^2 = 8x^5$ and $8x^3 * -6 = -48x^3$. This matches the original polynomial.
6. Final Answer
Therefore, the completely factored polynomial is $8x^3(x^2 - 6)$.
### 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 circuits. In finance, factoring can help simplify calculations when dealing with investments and compound interest. Understanding how to factor polynomials allows for efficient problem-solving in various fields.
- Find the GCF of the variable terms: GCF($x^5$, $x^3$) = $x^3$.
- Determine the overall GCF: $8x^3$.
- Factor out the GCF from the polynomial: $8x^5 - 48x^3 = 8x^3(x^2 - 6)$.
- The completely factored polynomial is $\boxed{8x^3(x^2 - 6)}$.
### Explanation
1. Understanding the Problem
We are asked to factor the polynomial $8x^5 - 48x^3$ completely using the greatest common factor (GCF). This means we need to identify the largest factor that divides both terms of the polynomial.
2. Finding the GCF of the Coefficients
First, let's find the GCF of the coefficients, which are 8 and -48. The factors of 8 are 1, 2, 4, and 8. The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. The greatest common factor of 8 and 48 is 8.
3. Finding the GCF of the Variable Terms
Next, let's find the GCF of the variable terms, which are $x^5$ and $x^3$. The GCF of $x^5$ and $x^3$ is $x^3$ because $x^3$ is the highest power of $x$ that divides both $x^5$ and $x^3$.
4. Determining the Overall GCF
Now, we combine the GCF of the coefficients and the GCF of the variable terms to find the overall GCF of the polynomial. The GCF is $8x^3$.
5. Factoring out the GCF
We factor out the GCF from the polynomial: $$8x^5 - 48x^3 = 8x^3(x^2 - 6)$$. To verify, we can distribute $8x^3$ back into the parentheses: $8x^3 * x^2 = 8x^5$ and $8x^3 * -6 = -48x^3$. This matches the original polynomial.
6. Final Answer
Therefore, the completely factored polynomial is $8x^3(x^2 - 6)$.
### 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 circuits. In finance, factoring can help simplify calculations when dealing with investments and compound interest. Understanding how to factor polynomials allows for efficient problem-solving in various fields.
