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

Mar 12, 2018

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:

$3 \left(17 {x}^{3} {y}^{2} - 9 x y + 23 y\right)$

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:

$3 y \left(17 {x}^{3} y - 9 x + 23\right)$

The greatest common factor is the value outside of the parentheses in the above equation, meaing your answer is $\textcolor{red}{3 y}$

Mar 12, 2018

$G C F \left(51 {x}^{3} {y}^{2} , - 27 x y , 69 y\right) = \textcolor{red}{3 y}$

#### Explanation:

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

$51 = \textcolor{b l u e}{3} \times 17$
$27 = \textcolor{b l u e}{3} \times 9$
$69 = \textcolor{b l u e}{3} \times 23$
$\textcolor{w h i t e}{\text{XXX}}$...by inspection $17 , 9 , \mathmr{and} 23$ have no common factors $> 1$

${x}^{3} {y}^{2} = \textcolor{m a \ge n t a}{y} \times {x}^{3} y$
$x y = \textcolor{m a \ge n t a}{y} \times x$
$y = \textcolor{m a \ge n t a}{y}$

Combining the factors: $\textcolor{b l u e}{3} \textcolor{m a \ge n t a}{y}$