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 largest factor that is common among the given monomials. This typically requires breaking down the monomials into their prime factors or using other methods to find the GCF.

Answer

The GCF of the monomials is $4x$.

Steps to Solve

Identify the monomials Start by clearly listing the given monomials. For example, let's say we have the monomials $12x^2$ and $8x$.
Factor each monomial Next, factor each monomial into its numerical and variable parts.

For $12x^2$, the numerical part factors as $12 = 2^2 \times 3$ and the variable part is $x^2$.
For $8x$, the numerical part factors as $8 = 2^3$ and the variable part is $x$.

Write the factors Write out the factored forms of the monomials:

$12x^2 = 2^2 \times 3 \times x^2$
$8x = 2^3 \times x$

Determine the GCF of the numerical coefficients Find the GCF of the numerical parts. The GCF of the coefficients $12$ and $8$ is found by taking the lowest powers of the common prime factors:

The prime factors are $2$: $min(2, 3) = 2$
Therefore, GCF of numerical coefficients is $2^2 = 4$.

Determine the GCF of the variable parts Now, find the GCF of the variable parts. Since $x^2$ and $x$ share the variable $x$, the GCF is $x^{min(2, 1)} = x^1 = x$.
Combine the GCFs Combine the GCF of the numerical and variable parts. The overall GCF is: $$ GCF = 4x $$.

The greatest common factor (GCF) of the monomials $12x^2$ and $8x$ is $4x$.

More Information

The greatest common factor is useful in simplifying fractions, factoring polynomials, and finding common terms in algebraic expressions. The GCF ensures that you can factor out the largest possible term from a given expression.

Tips

Forgetting to factor both the numerical and variable parts separately.
Taking the wrong minimum power for variable parts. Always ensure to compare the exponents correctly.

Thank you for voting!