Answer :
We are given the expressions:
[tex]$$1035 x^5 y^2,\quad 315 x^6,\quad 180 x^5,\quad 45 x^4.$$[/tex]
To find the greatest common factor (GCF), we work through the following steps:
1. Find the GCF of the coefficients:
The coefficients are 1035, 315, 180, and 45. The greatest common divisor of these numbers is 45.
2. Determine the common power of [tex]$x$[/tex]:
The exponents of [tex]$x$[/tex] in the terms are 5, 6, 5, and 4. The smallest exponent is 4, so the common factor contributed by [tex]$x$[/tex] is [tex]$x^4$[/tex].
3. Determine the common power of [tex]$y$[/tex]:
Only the first term has a factor involving [tex]$y$[/tex] (namely [tex]$y^2$[/tex]), while the others do not include [tex]$y$[/tex] (equivalent to [tex]$y^0$[/tex]). The smallest exponent among these is 0, meaning no [tex]$y$[/tex] factor is common to each term.
4. Combine the common factors:
Multiplying the common factors from the coefficients and the variable parts, we get:
[tex]$$45\cdot x^4 = 45 x^4.$$[/tex]
Thus, the greatest common factor of the given terms is:
[tex]$$45 x^4.$$[/tex]
This matches option A.
[tex]$$1035 x^5 y^2,\quad 315 x^6,\quad 180 x^5,\quad 45 x^4.$$[/tex]
To find the greatest common factor (GCF), we work through the following steps:
1. Find the GCF of the coefficients:
The coefficients are 1035, 315, 180, and 45. The greatest common divisor of these numbers is 45.
2. Determine the common power of [tex]$x$[/tex]:
The exponents of [tex]$x$[/tex] in the terms are 5, 6, 5, and 4. The smallest exponent is 4, so the common factor contributed by [tex]$x$[/tex] is [tex]$x^4$[/tex].
3. Determine the common power of [tex]$y$[/tex]:
Only the first term has a factor involving [tex]$y$[/tex] (namely [tex]$y^2$[/tex]), while the others do not include [tex]$y$[/tex] (equivalent to [tex]$y^0$[/tex]). The smallest exponent among these is 0, meaning no [tex]$y$[/tex] factor is common to each term.
4. Combine the common factors:
Multiplying the common factors from the coefficients and the variable parts, we get:
[tex]$$45\cdot x^4 = 45 x^4.$$[/tex]
Thus, the greatest common factor of the given terms is:
[tex]$$45 x^4.$$[/tex]
This matches option A.
