Identify the greatest common factor of $25 x^3 z^2$ and $50 c^3 y z^4$.

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01-11-2026
Mathematics

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Identify the greatest common factor of $25 x^3 z^2$ and $50 c^3 y z^4$.

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-01T22:09:34-05:00

Identify numerical coefficients: 25 and 50, with GCF 25.
Identify common variables: Only $z$ is shared.
Find minimum exponent of common variables: $\min(2, 4) = 2$, resulting in $z^2$.
Multiply components: The GCF is $\boxed{25z^2}$.

Explanation

Problem Analysis

We are given two algebraic expressions (monomials): $25 x^3 z^2$ and $50 c^3 y z^4$. Our goal is to find the greatest common factor (GCF) of these two expressions. The GCF of monomials is defined as the product of the GCF of their numerical coefficients and the lowest power of each common variable present in both expressions. We will analyze the coefficients and the variables separately.

GCF of Numerical Coefficients

First, identify the numerical coefficients, which are 25 and 50. We need to find the GCF of 25 and 50. Since $25 = 25 \times 1$ and $50 = 25 \times 2$, the greatest common divisor is clearly 25.

Variable Analysis and Final GCF Construction

Next, examine the variables present in both monomials. The first monomial, $25 x^3 z^2$, contains the variables $x$ and $z$. The second monomial, $50 c^3 y z^4$, contains the variables $c$, $y$, and $z$. Comparing these, the only variable common to both expressions is $z$. The exponent of $z$ in the first expression is 2, and in the second expression, it is 4. The GCF uses the lowest power of the common variable found in both, so we take the minimum of 2 and 4, which is 2. Therefore, the common variable factor is $z^2$.

Conclusion

Finally, combine the GCF of the coefficients and the variable factor. Multiplying the coefficient GCF (25) by the variable part ($z^2$), we get the greatest common factor for the entire expression. Thus, the GCF of $25 x^3 z^2$ and $50 c^3 y z^4$ is $\boxed{25z^2}$.

Examples

The concept of the greatest common factor (GCF) is frequently used in simplifying algebraic fractions. For instance, if you were to simplify a fraction like $\frac{25 x^3 z^2}{50 c^3 y z^4}$, you would first identify the GCF to divide both the numerator and denominator by it. This results in $\frac{25 z^2 \cdot x^3}{25 z^2 \cdot 2 c^3 y} = \frac{x^3}{2 c^3 y}$. This technique is essential for algebraic manipulation and solving equations efficiently, similar to how finding a common factor helps in reducing physical measurements or resource distribution in real-world engineering or design projects.