Identify the greatest common factor of $28 b c^3 y^3$ and $8 a y^4$.

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aarilyn7 aarilyn7

01-11-2026
Mathematics

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Identify the greatest common factor of $28 b c^3 y^3$ and $8 a y^4$.

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-01T22:07:23-05:00

Identify numerical coefficients: 28 and 8. Their GCF is 4.
Identify common variables: Only $y$ appears in both monomials.
Compare exponents for common variables: The powers of $y$ are $y^3$ and $y^4$. The lowest exponent is 3, so the common factor is $y^3$.
Combine results: The greatest common factor is $4y^3$.
Final Answer: $\boxed{4y^3}$.

Explanation

Analysis of the Monomials

To find the greatest common factor (GCF) of two monomials, we need to consider the numerical coefficients and the literal parts (variables and their exponents) separately. The two monomials given are $28bc^3y^3$ and $8ay^4$. The GCF will be the product of the GCF of the numerical coefficients and the greatest common power of each variable present in both terms.

Finding the GCF of the Coefficients

First, we determine the GCF of the numerical coefficients, which are 28 and 8.

The prime factorization of 28 is $2^2 \times 7 = 4 \times 7$.
The prime factorization of 8 is $2^3 = 8$.

The common prime factor is 2, and the lowest exponent is 2. Thus, the GCF of 28 and 8 is $2^2 = 4$.

Finding the GCF of the Literal Parts

Next, we examine the variables in both monomials to see which ones are common to both:

Variable $a$: appears in $8ay^4$ but not in $28bc^3y^3$. Since it is not in both, it does not contribute to the GCF.
Variable $b$: appears in $28bc^3y^3$ but not in $8ay^4$. Since it is not in both, it does not contribute to the GCF.
Variable $c$: appears in $28bc^3y^3$ but not in $8ay^4$. Since it is not in both, it does not contribute to the GCF.
Variable $y$: appears in both $28bc^3y^3$ and $8ay^4$. The exponents are 3 and 4, respectively. The lowest exponent is 3. Therefore, $y^3$ is a common factor.

Final Calculation of the GCF

Combining the GCF of the coefficients and the GCF of the variable parts, we multiply them together:
GCF = (GCF of coefficients) $\times$ (common variables)
GCF = $4 \times y^3 = 4y^3$.
Thus, the greatest common factor of $28bc^3y^3$ and $8ay^4$ is $4y^3$.

Examples

Finding the greatest common factor is essential in algebra for simplifying expressions, such as factoring polynomials. For example, if you are designing a storage space and have two different items with dimensions described by monomials, finding the GCF helps determine the size of the standard containers required to hold both types of items compactly. This skill is foundational for solving algebraic equations and managing resources efficiently.