How to find the GCF of monomials?
Understand the Problem
The question is asking how to determine the greatest common factor (GCF) of monomials, which involves identifying the highest factor shared among the given monomials. We can approach this by factoring each monomial and finding the common factors.
Answer
The GCF of the monomials is $6x^2$.
Answer for screen readers
The greatest common factor (GCF) of the monomials $12x^2$ and $18x^3$ is $6x^2$.
Steps to Solve
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Identify the Monomials First, list out the given monomials. For example, if the monomials are $12x^2$ and $18x^3$, we will work with these.
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Factor the Numerical Coefficients Next, find the factors of the numerical parts of the monomials:
- For $12 = 2^2 \cdot 3^1$
- For $18 = 2^1 \cdot 3^2$
- Identify Common Factors in Coefficients Look for the common factors in the numerical coefficients:
- The common factors are $2$ and $3$.
- The lowest power of $2$ is $2^1$ and the lowest power of $3$ is $3^1$.
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Calculate the GCF of Numerical Coefficients Now, multiply the common factors together: $$ \text{GCF of coefficients} = 2^1 \cdot 3^1 = 6 $$
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Identify the Variable Part Next, examine the variables in the monomials:
- From $12x^2$, you have $x^2$.
- From $18x^3$, you have $x^3$.
- Determine the Lowest Power of Each Variable Identify the lowest power of each variable:
- For the variable $x$, the lowest power is $x^2$.
- Combine GCF of Coefficients and Variables Now, combine the GCF of the numerical coefficients with the GCF of the variables: $$ \text{GCF} = 6x^2 $$
The greatest common factor (GCF) of the monomials $12x^2$ and $18x^3$ is $6x^2$.
More Information
The GCF of monomials helps to solve problems in algebra, such as simplifying expressions or factoring polynomials. It also plays a crucial role in finding common denominators and reducing fractions.
Tips
- Forgetting to find the lowest powers of variables. Always ensure to check each variable and select the lowest exponent.
- Neglecting to factor both the coefficients and the variables. It’s important to treat both parts separately before combining them.
