What is the GCF of $28 w x^4 z^3$ and $42 w^4 x^2 z^5$?

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06-11-2026
Mathematics

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What is the GCF of $28 w x^4 z^3$ and $42 w^4 x^2 z^5$?

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-06T12:19:10-05:00

Find the GCF of the coefficients $28$ and $42$, which is $14$.
Determine the minimum exponent for each variable $w, x, z$ present in both monomials: $w^{\min(1, 4)} = w^1$, $x^{\min(4, 2)} = x^2$, and $z^{\min(3, 5)} = z^3$.
Multiply the coefficient GCF by the literal GCF to obtain the final result.
The greatest common factor is $\boxed{14wx^2z^3}$.

Explanation

Problem Analysis and Strategy

To find the Greatest Common Factor (GCF) of the two monomials $28w x^4 z^3$ and $42w^4 x^2 z^5$, we separate the problem into two parts: the numerical coefficients and the literal parts (variables with their exponents). The GCF will be the product of the GCF of the coefficients and the GCF of the literal parts.

Finding the GCF of the Numerical Coefficients

The numerical coefficients are 28 and 42. We find their GCF by prime factorization:

For 28: $28 = 2^2 \times 7$
For 42: $42 = 2 \times 3 \times 7$

The common prime factors are 2 and 7, each taken to the minimum power present in both factorizations. Thus, $\text{GCF}(28, 42) = 2^1 \times 7^1 = 14$.

Finding the GCF of the Literal Parts

The literal parts are $w^1 x^4 z^3$ and $w^4 x^2 z^5$. To find the GCF of these terms, we take each variable common to both monomials raised to the smallest exponent found in either monomial:

For $w$: $\min(1, 4) = 1$, so we take $w^1$.
For $x$: $\min(4, 2) = 2$, so we take $x^2$.
For $z$: $\min(3, 5) = 3$, so we take $z^3$.

Multiplying these together, the GCF of the variable parts is $w^1 x^2 z^3$.

Combining Results to Finalize the GCF

Finally, we combine the numerical GCF and the literal GCF to find the GCF of the original monomials:
\[ \text{GCF}(28w x^4 z^3, 42w^4 x^2 z^5) = 14 \cdot (w x^2 z^3) = 14w x^2 z^3 \]
The greatest common factor of the given monomials is $\boxed{14wx^2z^3}$.

Examples

Finding the greatest common factor of monomials is a fundamental skill used in simplifying algebraic expressions and factoring polynomials. For example, if you have a manufacturing process where two types of products are bundled in packages described by these monomials, the GCF represents the largest number of identical sets of items you can extract from both bundles. This ensures efficiency in logistics and inventory management, allowing for organized distribution or reduction of complex expressions into a more manageable form.