Exercise 56 Page 51

We want to find the greatest common factor (GCF) of the terms in the given expression. To do so, we will consider coefficients and both variables separately. 2 m^3p^2- 16 mp^2+ 8 mp Let's start by finding the GCF of 2, 16 and 8. Factors of2:& 1 and 2 Factors of16:& 1, 2,4,8 and 16 Factors of8:& 1, 2,4 and 8 We found that the GCF of the coefficients is 2. To find the GCF of the variables, we need to identify the variables repeated in every term, and write them with their minimum exponents. Let's start with the variable m. \begin{aligned} \textbf{Factors of }\bm{1^\text{st}}\textbf{ Variable:}&\ {\color{#FF0000}{m}}, m^2, m^3\\ \textbf{Factors of }\bm{2^\text{nd}}\textbf{ Variable:}&\ {\color{#FF0000}{m}} \\ \textbf{Factors of }\bm{3^\text{rd}}\textbf{ Variable:}&\ {\color{#FF0000}{m}} \end{aligned} We see that there is one repeated variable factor, m. Now, let's move on to the variable p. \begin{aligned} \textbf{Factors of }\bm{1^\text{st}}\textbf{ Variable:}&\ {\color{#FF0000}{p}}, p^2\\ \textbf{Factors of }\bm{2^\text{nd}}\textbf{ Variable:}&\ {\color{#FF0000}{p}}, p^2 \\ \textbf{Factors of }\bm{3^\text{rd}}\textbf{ Variable:}&\ {\color{#FF0000}{p}} \end{aligned} Also, only one term, p, is repeated. Therefore, the GCF of the expression is 2* m* p= 2mp. Now we can write the given expression in terms of the GCF. 2m^3p^2-16mp^2+8mp ⇕ 2mp* m^2p- 2mp* 8p+ 2mp * 4 Finally, we will factor out the GCF. 2mp* m^2p- 2mp* 8p+ 2mp * 4 ⇕ 2mp(m^2p-8p+4)