Certainly! Let's find the Greatest Common Monomial Factor (GCMF) of the polynomials [tex]\(x^2\)[/tex] and [tex]\(2x\)[/tex]. Here is a detailed step-by-step process:
1. Identify the given polynomials:
We are given the polynomials [tex]\(x^2\)[/tex] and [tex]\(2x\)[/tex].
2. Express each polynomial in terms of its monomials:
- The first polynomial is [tex]\(x^2\)[/tex]. This monomial includes the factor [tex]\(x\)[/tex] raised to the power of 2.
- The second polynomial is [tex]\(2x\)[/tex]. This monomial includes the factor [tex]\(x\)[/tex] raised to the power of 1, multiplied by the coefficient 2.
3. Find the GCMF of the monomials:
- To determine the GCMF, we need to consider the variables and their powers in each monomial.
- The greatest common monomial factor must have the highest power of [tex]\(x\)[/tex] that is common to both polynomials.
- The powers of [tex]\(x\)[/tex] in the given polynomials are 2 (from [tex]\(x^2\)[/tex]) and 1 (from [tex]\(2x\)[/tex]). The smallest power of [tex]\(x\)[/tex] that is common to both is 1.
4. Determine the common factor:
- The common factor for the variable [tex]\(x\)[/tex] is [tex]\(x^1\)[/tex] or simply [tex]\(x\)[/tex].
Therefore, the Greatest Common Monomial Factor (GCMF) of the polynomials [tex]\(x^2\)[/tex] and [tex]\(2x\)[/tex] is [tex]\(\boxed{x}\)[/tex].
