Answer :
To find the greatest common monomial factor of the expression [tex]\(6m^3x^4 + 30m^5y^2 + 45m^3x^6y^2\)[/tex], we need to look at each part of the terms:
1. Coefficients:
- The coefficients of the terms are 6, 30, and 45.
- To find the greatest common divisor (GCD) of these coefficients, consider:
- 6 = [tex]\(2 \times 3\)[/tex]
- 30 = [tex]\(2 \times 3 \times 5\)[/tex]
- 45 = [tex]\(3 \times 3 \times 5\)[/tex]
- The GCD of these coefficients is 3, since 3 is the greatest number that divides all the coefficients evenly.
2. Variable [tex]\(m\)[/tex]:
- The powers of [tex]\(m\)[/tex] in the terms are 3, 5, and 3.
- The smallest power of [tex]\(m\)[/tex] that appears in all terms is 3.
- So, the factor for [tex]\(m\)[/tex] is [tex]\(m^3\)[/tex].
3. Variable [tex]\(x\)[/tex]:
- The powers of [tex]\(x\)[/tex] in the terms are 4, 0, and 6.
- Since one term does not have [tex]\(x\)[/tex] (power 0), [tex]\(x\)[/tex] is not a common factor across all terms.
4. Variable [tex]\(y\)[/tex]:
- The powers of [tex]\(y\)[/tex] are 0, 2, and 2.
- The smallest power that [tex]\(y\)[/tex] appears with in every term is 0, meaning it’s not a factor across all terms.
So, combining these, the greatest common monomial factor of the expression is [tex]\(3m^3\)[/tex]. This monomial can be factored out from each term of the original expression.
1. Coefficients:
- The coefficients of the terms are 6, 30, and 45.
- To find the greatest common divisor (GCD) of these coefficients, consider:
- 6 = [tex]\(2 \times 3\)[/tex]
- 30 = [tex]\(2 \times 3 \times 5\)[/tex]
- 45 = [tex]\(3 \times 3 \times 5\)[/tex]
- The GCD of these coefficients is 3, since 3 is the greatest number that divides all the coefficients evenly.
2. Variable [tex]\(m\)[/tex]:
- The powers of [tex]\(m\)[/tex] in the terms are 3, 5, and 3.
- The smallest power of [tex]\(m\)[/tex] that appears in all terms is 3.
- So, the factor for [tex]\(m\)[/tex] is [tex]\(m^3\)[/tex].
3. Variable [tex]\(x\)[/tex]:
- The powers of [tex]\(x\)[/tex] in the terms are 4, 0, and 6.
- Since one term does not have [tex]\(x\)[/tex] (power 0), [tex]\(x\)[/tex] is not a common factor across all terms.
4. Variable [tex]\(y\)[/tex]:
- The powers of [tex]\(y\)[/tex] are 0, 2, and 2.
- The smallest power that [tex]\(y\)[/tex] appears with in every term is 0, meaning it’s not a factor across all terms.
So, combining these, the greatest common monomial factor of the expression is [tex]\(3m^3\)[/tex]. This monomial can be factored out from each term of the original expression.
