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Answer :
- Find the prime factorization of the coefficients 45 and 40, and identify the common factors.
- Determine the lowest power of x present in both terms.
- Multiply the common factors of the coefficients and the lowest power of x to find the GCF.
- The GCF of $45x^3 - 40x^2$ is $\boxed{5x^2}$.
### Explanation
1. Understanding the Problem
We are asked to find the greatest common factor (GCF) of the expression $45x^3 - 40x^2$. The GCF is the largest expression that divides evenly into both terms.
2. Finding the GCF of the Coefficients
First, let's find the GCF of the coefficients, 45 and 40. The prime factorization of 45 is $3^2 \times 5$, and the prime factorization of 40 is $2^3 \times 5$. The only common prime factor is 5, which appears with a power of 1 in both factorizations. Therefore, the GCF of 45 and 40 is 5.
3. Finding the GCF of the Variables
Next, let's find the GCF of the variable parts, $x^3$ and $x^2$. The GCF is the lowest power of $x$ that appears in both terms, which is $x^2$.
4. Combining the GCFs
Finally, we multiply the GCF of the coefficients and the GCF of the variables to find the GCF of the entire expression. This gives us $5x^2$.
5. Final Answer
Therefore, the GCF of $45x^3 - 40x^2$ is $5x^2$.
### Examples
Understanding the greatest common factor is useful in many real-life situations, such as when you're trying to divide a garden into equal sections. For example, if you have a rectangular garden that is 45 feet long and 40 feet wide, the greatest common factor (5) tells you that you can divide both the length and the width into sections of 5 feet each to create equal-sized square plots. This concept is also used in simplifying fractions, optimizing designs, and scheduling tasks.
- Determine the lowest power of x present in both terms.
- Multiply the common factors of the coefficients and the lowest power of x to find the GCF.
- The GCF of $45x^3 - 40x^2$ is $\boxed{5x^2}$.
### Explanation
1. Understanding the Problem
We are asked to find the greatest common factor (GCF) of the expression $45x^3 - 40x^2$. The GCF is the largest expression that divides evenly into both terms.
2. Finding the GCF of the Coefficients
First, let's find the GCF of the coefficients, 45 and 40. The prime factorization of 45 is $3^2 \times 5$, and the prime factorization of 40 is $2^3 \times 5$. The only common prime factor is 5, which appears with a power of 1 in both factorizations. Therefore, the GCF of 45 and 40 is 5.
3. Finding the GCF of the Variables
Next, let's find the GCF of the variable parts, $x^3$ and $x^2$. The GCF is the lowest power of $x$ that appears in both terms, which is $x^2$.
4. Combining the GCFs
Finally, we multiply the GCF of the coefficients and the GCF of the variables to find the GCF of the entire expression. This gives us $5x^2$.
5. Final Answer
Therefore, the GCF of $45x^3 - 40x^2$ is $5x^2$.
### Examples
Understanding the greatest common factor is useful in many real-life situations, such as when you're trying to divide a garden into equal sections. For example, if you have a rectangular garden that is 45 feet long and 40 feet wide, the greatest common factor (5) tells you that you can divide both the length and the width into sections of 5 feet each to create equal-sized square plots. This concept is also used in simplifying fractions, optimizing designs, and scheduling tasks.
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