Greatest Common Factor: Find GCF Of 30u⁵v⁷ And 18u⁴v³w⁸

Have you ever wondered how to find the greatest common factor (GCF) of algebraic expressions? It might seem daunting at first, but with a systematic approach, it becomes quite manageable. In this guide, we'll break down the process using the expressions $30 u^5 v^7$ and $18 u^4 v^3 w^8$ as our examples. Understanding the GCF is crucial in simplifying expressions, solving equations, and various other mathematical operations. So, let’s dive in and unravel this concept together.

1. Understanding the Greatest Common Factor (GCF)

In mathematics, the greatest common factor (GCF), also known as the greatest common divisor (GCD), is the largest positive integer that divides two or more integers without a remainder. When dealing with algebraic expressions, the GCF includes both the largest numerical factor and the highest powers of common variables. Finding the GCF is an essential skill in algebra, as it helps in simplifying expressions, factoring polynomials, and solving equations. The GCF is the largest expression that can divide evenly into all the given terms. Think of it as the biggest piece you can pull out of each term, leaving behind simpler expressions. This is particularly useful when you're trying to simplify fractions or factor polynomials. For instance, if you have the expression $6x^2 + 9x$, finding the GCF (which is $3x$) allows you to rewrite the expression as $3x(2x + 3)$, which is a simpler, factored form.

2. Prime Factorization of Coefficients

To find the GCF, the first step involves breaking down the coefficients (the numerical parts) into their prime factors. Prime factorization is the process of expressing a number as a product of its prime factors, which are numbers that are only divisible by 1 and themselves. Let's apply this to our coefficients, 30 and 18. For 30, the prime factorization is $2 imes 3 imes 5$. This means that 30 can be written as the product of the prime numbers 2, 3, and 5. Similarly, for 18, the prime factorization is $2 imes 3 imes 3$, or $2 imes 3^2$. Breaking down the numbers into their prime factors makes it easier to identify common factors. By expressing the coefficients in terms of their prime factors, we can clearly see which factors are shared between the numbers, which is a crucial step in determining the GCF. This method ensures that we don't miss any common factors, and it provides a solid foundation for the subsequent steps in finding the GCF of the given expressions.

3. Identifying Common Variable Factors

Next, we need to identify the variables that are common to both expressions. In our example, we have the variables $u$, $v$, and $w$. However, notice that $w$ appears only in the second expression ($18 u^4 v^3 w^8$). Therefore, $w$ is not a common variable and will not be part of the GCF. The common variables are $u$ and $v$. Now, we determine the lowest power of each common variable present in both expressions. For $u$, the powers are $u^5$ and $u^4$. The lowest power is $u^4$. For $v$, the powers are $v^7$ and $v^3$. The lowest power is $v^3$. Remember, the GCF includes the lowest power of the common variables because that's the highest power that can divide evenly into both terms. For example, $u^5$ cannot divide evenly into $u^4$ without leaving a remainder, but $u^4$ can divide evenly into both $u^5$ and $u^4$. Similarly, $v^7$ cannot divide evenly into $v^3$, but $v^3$ can divide evenly into both $v^7$ and $v^3$. This careful selection of the lowest powers ensures that the GCF we find is indeed the greatest common factor.

4. Combining the Factors to Find the GCF

Now that we have the prime factors of the coefficients and the lowest powers of the common variables, we can combine them to find the GCF. From the prime factorization, we identified that both 30 ($2 imes 3 imes 5$) and 18 ($2 imes 3^2$) share the prime factors 2 and 3. The lowest power of 2 that appears in both factorizations is $2^1$ (or simply 2), and the lowest power of 3 that appears in both is $3^1$ (or simply 3). Therefore, the numerical part of the GCF is $2 imes 3 = 6$. We also found that the lowest powers of the common variables are $u^4$ and $v^3$. Combining the numerical factor with the variable factors, we get the GCF as $6 u^4 v^3$. This means that $6 u^4 v^3$ is the largest expression that can divide evenly into both $30 u^5 v^7$ and $18 u^4 v^3 w^8$. To verify, you can divide both original expressions by the GCF to see if you get simpler expressions without any common factors remaining. This step-by-step process ensures that we accurately determine the GCF by considering both the numerical coefficients and the variable components of the given expressions.

5. Verification and Final Answer

To ensure we've found the correct GCF, we can divide the original expressions by our calculated GCF, $6u^4v^3$. Let's do that:

• For $30u^5v^7$, dividing by $6u^4v^3$ gives us $(30/6) imes (u^5/u^4) imes (v^7/v^3) = 5uv^4$.
• For $18u^4v^3w^8$, dividing by $6u^4v^3$ gives us $(18/6) imes (u^4/u^4) imes (v^3/v^3) imes w^8 = 3w^8$.

Since $5uv^4$ and $3w^8$ have no common factors other than 1, our GCF is indeed correct. Therefore, the greatest common factor of $30 u^5 v^7$ and $18 u^4 v^3 w^8$ is $6 u^4 v^3$. This final verification step is crucial because it confirms that we haven't missed any common factors and that the GCF we've found is the largest possible factor that divides both expressions. It provides a sense of assurance and completeness to the solution. Additionally, understanding how to verify the GCF reinforces the underlying concepts and helps in developing a deeper understanding of algebraic manipulations.

Conclusion

Finding the greatest common factor involves breaking down expressions into their prime factors and identifying common elements. By following these steps—prime factorization of coefficients, identifying common variable factors, and combining them—you can confidently find the GCF of any algebraic expressions. This skill is invaluable in simplifying expressions and solving various mathematical problems. Remember, practice makes perfect, so keep applying these steps to different problems to master the technique. Understanding the GCF is not just about finding an answer; it's about developing a deeper understanding of the structure of mathematical expressions and how they relate to each other. This understanding will serve you well as you continue your journey in mathematics.

For further learning and practice on finding the greatest common factor, you can visit websites like Khan Academy, which offers comprehensive lessons and exercises on this topic.