Exercise 34 Page 174

We want to find the greatest common factor — GCF — of the terms in the given expression. To do so, we will consider coefficients and variables separately. 18 x^2y^2- 24 xy^2+ 36 y^2 Let's start by finding the GCF of 18, 24, and 36. Factors of18:& 1,2,3, 6,9,and18 Factors of24:& 1,2,3,4, 6,8,12,and24 Factors of36:& 1,2,3,4, 6,9,12,18and 36 We found that the GCF of the coefficients is 6. To find the GCF of the variables, we need to identify the variables repeated in both terms, and write them with their minimum exponents. \begin{aligned} \textbf{Factors of }& \bm{1^\text{st}}\textbf{ Variable:}\ x, x^2, y, {\color{#FF0000}{y^2}}, xy, x^2y, xy^2, x^2y^2\\ \textbf{Factors of }& \bm{2^\text{nd}}\textbf{ Variable:}\ x, y, {\color{#FF0000}{y^2}}, xy, xy^2 \\ \textbf{Factors of }& \bm{3^\text{rd}}\textbf{ Variable:}\ y, {\color{#FF0000}{y^2}} \end{aligned} We see that there is one repeated variable factor, y^2. Thus, the GCF of the expression is 6* y^2= 6y^2. Now we can write the given expression in terms of the GCF. 18x^2y^2-24xy^2+36y^2 ⇕ 6y^2* 3x^2- 6y^2* 4x + 6y^2* 6 Finally, we will factor out the GCF. 6y^2* 3x^2- 6y^2* 4x+ 6y^2* 6 ⇕ 6y^2(3x^2-4x+6)