Identify the greatest common factor of $11 b^2 x^2$ and $11 c^4 z^4$.

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01-11-2026
Mathematics

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Identify the greatest common factor of $11 b^2 x^2$ and $11 c^4 z^4$.

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-01T22:08:56-05:00

Identify the numerical coefficients: both expressions have a coefficient of 11, so the GCF of the coefficients is 11.
Analyze the variables: the first expression contains $b$ and $x$, while the second contains $c$ and $z$.
Find common variables: there are no common variables between the two expressions.
Combine the results: the GCF is the product of the numerical GCF and any common variable powers, resulting in $\boxed{11}$.

Explanation

Analyzing the Expressions and the GCF Concept

To determine the greatest common factor (GCF) of the two algebraic expressions $11b^2x^2$ and $11c^4z^4$, we first examine their numerical coefficients and variable components. The GCF of two expressions is the largest expression that divides both original expressions without a remainder. It is calculated by finding the GCF of the numerical coefficients and multiplying it by the smallest power of each common variable present in both expressions.

Finding the GCF of the Numerical Coefficients

The numerical coefficients in both expressions $11b^2x^2$ and $11c^4z^4$ are both 11. The greatest common divisor of 11 and 11 is simply 11, as 11 is a prime number and the largest integer that divides both is the number itself. Thus, the GCF of the numerical coefficients is 11.

Identifying Common Variables

Next, we identify the variables present in each expression. The first expression, $11b^2x^2$, contains the variables $b$ and $x$. The second expression, $11c^4z^4$, contains the variables $c$ and $z$. Comparing these, we look for any variables that appear in both sets. Since there are no variables common to both expressions (i.e., the set of variables $\{b, x\}$ and $\{c, z\}$ have an empty intersection), the product of the common variables raised to their lowest powers is effectively 1.

Conclusion

Finally, we combine the GCF of the numerical coefficients with the product of the common variable parts. Since the GCF of the coefficients is 11 and there are no common variables to include, the greatest common factor of the entire expressions is 11 multiplied by 1, which equals 11. Therefore, the greatest common factor of $11b^2x^2$ and $11c^4z^4$ is $\boxed{11}$.

Examples

Understanding the greatest common factor is like finding the largest size of a container that can divide two different quantities of items perfectly. For instance, if you have 11 blue marbles and 22 red marbles, the GCF helps you distribute these into the largest possible groups of the same size without any leftovers. In algebra, finding the GCF of expressions like $11b^2x^2$ and $11c^4z^4$ allows us to simplify complex fractions, solve equations, or factor out common terms, making mathematical manipulations much more manageable and efficient.