Let's solve the given problems step by step to find the Greatest Common Monomial Factor (GCMF) for each pair of expressions.
### 1. Finding the GCMF of [tex]\( x^2 \)[/tex] and [tex]\( 2x \)[/tex]
To find the GCMF of [tex]\( x^2 \)[/tex] and [tex]\( 2x \)[/tex], we need to find the highest degree of [tex]\( x \)[/tex] that divides both expressions.
- [tex]\( x^2 \)[/tex] can be factored as [tex]\( x \cdot x \)[/tex]
- [tex]\( 2x \)[/tex] can be written as [tex]\( 2 \cdot x \)[/tex]
The common factor between these expressions is [tex]\( x \)[/tex].
Thus, the GCMF of [tex]\( x^2 \)[/tex] and [tex]\( 2x \)[/tex] is:
[tex]\[ x \][/tex]
### 2. Finding the GCMF of [tex]\( 4x^3 \)[/tex] and [tex]\( 8x^2 \)[/tex]
To find the GCMF of [tex]\( 4x^3 \)[/tex] and [tex]\( 8x^2 \)[/tex], we need to consider both the coefficients and the powers of [tex]\( x \)[/tex].
- [tex]\( 4x^3 \)[/tex] can be factored as [tex]\( 4 \cdot x \cdot x \cdot x \)[/tex]
- [tex]\( 8x^2 \)[/tex] can be factored as [tex]\( 8 \cdot x \cdot x \)[/tex]
First, we find the GCD of the coefficients:
[tex]\[ \text{GCD of } 4 \text{ and } 8 \text{ is } 4 \][/tex]
Next, we find the lowest power of [tex]\( x \)[/tex] common to both expressions:
[tex]\[ x^3 \text{ and } x^2 \text{ have a common factor of } x^2 \][/tex]
Combining these, the GCMF of [tex]\( 4x^3 \)[/tex] and [tex]\( 8x^2 \)[/tex] is:
[tex]\[ 4x^2 \][/tex]
### 3. Finding the GCMF of [tex]\( 15y \)[/tex] and 92
To find the GCMF of [tex]\( 15y \)[/tex] and 92, we need to consider the coefficients since the second term (92) has no variables.
- [tex]\( 15y \)[/tex] can be written as [tex]\( 15 \cdot y \)[/tex]
- [tex]\( 92 \)[/tex] remains as [tex]\( 92 \)[/tex]
GCMF in terms of the coefficients only:
[tex]\[ \text{GCD of } 15 \text{ and } 92 \text{ is } 1 \][/tex]
Since there’s no common variable part and the GCD is 1, the GCMF of [tex]\( 15y \)[/tex] and 92 is:
[tex]\[ 1 \][/tex]
### Summary of Results
Combining the results for each part, we have:
1. GCMF of [tex]\( x^2 \)[/tex] and [tex]\( 2x \)[/tex] is: [tex]\( x \)[/tex]
2. GCMF of [tex]\( 4x^3 \)[/tex] and [tex]\( 8x^2 \)[/tex] is: [tex]\( 4x^2 \)[/tex]
3. GCMF of [tex]\( 15y \)[/tex] and 92 is: [tex]\( 1 \)[/tex]
So the final combined results are:
[tex]\[ (x, 4x^2, 1) \][/tex]
