What is the greatest common factor of #51x^3y^2 - 27xy + 69y#?

2 Answers
Mar 12, 2018

Answer:

3y

Explanation:

I did this in two steps. I first looked at the numeric coefficients to determine if there was a common factor for the polynomial:

51 -27 69

51 is divisible by 3 and 17
27 is divisible by 3 and 9, and 9 is #3^2#, meaning #27 = 3^3#
69 is divisible by 3 and 23

since the shared factor among the three coefficients is 3, we can pull that out of the whole equation as a common factor:

#3(17x^3y^2-9xy+23y)#

Next, we can see if there are non-numeric coefficients (x and y in this case) that are used in all 3 terms. x is used twice, but y is found in all three terms. This means we can pull y out of the equation. You do this by dividing all 3 terms by y and putting a y outside the parentheses:

#3y(17x^3y-9x+23)#

The greatest common factor is the value outside of the parentheses in the above equation, meaing your answer is #color(red)(3y)#

Mar 12, 2018

Answer:

#GCF(51x^3y^2,-27xy,69y)=color(red)(3y)#

Explanation:

Find the GCF of the constants and the composite variables separately:

#51=color(blue)3xx17#
#27=color(blue)3xx9#
#69=color(blue)3xx23#
#color(white)("XXX")#...by inspection #17,9, and 23# have no common factors #>1#

#x^3y^2=color(magenta)yxx x^3y#
#xy=color(magenta)y xx x#
#y=color(magenta)y#

Combining the factors: #color(blue)3color(magenta)y#