How to find greatest common factor with variables?
Understand the Problem
The question is asking how to find the greatest common factor (GCF) when dealing with algebraic variables. This typically involves identifying the highest degree of common terms or coefficients present in the given variables.
Answer
The GCF is $4xy$.
Answer for screen readers
The greatest common factor (GCF) is $4xy$.
Steps to Solve
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Identify the Terms List the algebraic expressions for which you need to find the GCF. For example, consider the expressions $12x^2y$ and $16xy^2$.
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Factor the Coefficients Break down the coefficients into their prime factors. For $12$, the prime factorization is: $$ 12 = 2^2 \times 3 $$ For $16$, the prime factorization is: $$ 16 = 2^4 $$
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Determine the Common Factors of the Coefficients Identify the lowest power of common prime factors. Here, the common factor is $2$, and the minimum power is $2$: $$ \text{GCF of coefficients} = 2^2 = 4 $$
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Identify the Variables For the variables $x^2y$ and $xy^2$, find the lowest exponent for each variable present in both terms:
- For $x$: The lowest exponent is $1$ (from $xy$).
- For $y$: The lowest exponent is $1$ (from $xy$).
- Combine the Results Now, combine the GCF of the coefficients with the variables: $$ \text{GCF} = 4x^1y^1 = 4xy $$
The greatest common factor (GCF) is $4xy$.
More Information
The GCF is important in simplifying fractions and expressions in algebra. It helps in factoring polynomials and finding common terms in expressions, which can simplify problems significantly.
Tips
- Forgetting to consider the variables along with the coefficients when finding the GCF.
- Not properly factoring the coefficients, leading to incorrect identification of common factors.
