How to find the GCF of a monomial?

Understand the Problem

The question is asking how to calculate the greatest common factor (GCF) of a monomial, which involves identifying the largest monomial that divides a given monomial without leaving a remainder.

Answer

The GCF is the product of the GCF of coefficients and the lowest powers of the variables.

Steps to Solve

Identify the coefficients and variables of the monomial

Start by looking at the monomial given in the problem. Separate the numerical coefficients and the variables. For example, if the monomial is $12x^3y^2$, the coefficient is $12$, and the variables are $x^3$ and $y^2$.

Find the GCF of the coefficients

Next, identify the coefficients of any other monomials you're comparing to find the GCF. Use prime factorization to find the GCF of the numerical coefficients. For the coefficient $12$, the prime factorization is:

$$ 12 = 2^2 \times 3^1 $$

Identify the lowest power of each variable

Now examine the variables in the monomial. For each variable present, take the lowest exponent appearing in the compared monomials. For example, if you have $x^3$ and $x^2$, the lowest power is $x^2$.

Combine the GCF of coefficients and variables

After you have the GCF of the coefficients and the lowest powers of the variables, multiply them together to find the GCF. If the GCF of the coefficient is $6$ and the lowest powers of the variables are $x^2$ and $y^1$, then:

$$ \text{GCF} = 6x^2y $$

The greatest common factor (GCF) of the given monomial and any other compared monomials will be a product of the GCF of the coefficients and the lowest powers of the variables.

More Information

Finding the GCF of monomials is useful in simplifying expressions and factoring polynomials. It reflects the largest shared factor among the given terms.

Tips

Forgetting to use prime factorization for coefficients can lead to incorrect GCF.
Not considering all variables and their respective powers properly.

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