Exercise 56 Page 341

We want to rewrite the given sum expression as an equivalent expression that is a product. For example, 13x-13 is a sum while 13(x-1) is a product. Let's look at our sum expression! 32x^2+48x We can start by finding the greatest common factor (GCF) of the terms in the given expression. To do so, we will consider coefficients and variables separately. 32 x^2+ 48 x Let's start by finding the GCF of 32 and 48 by listing all possible factors of both numbers. Factors of32:& 1,2,4,8, 16, and32 Factors of48:& 1,2,3,4,6,8,12, & 16,24, and 48 We found that the GCF of the coefficients is 16. To find the GCF of the variables, we need to identify the all possible variable factors in both terms. \begin{aligned} \textbf{Factors of }\bm{1^\textbf{st}}\textbf{ Variable:}&\ {\color{#FF0000}{x}}, x^2\\ \textbf{Factors of }\bm{2^\textbf{nd}}\textbf{ Variable:}&\ {\color{#FF0000}{x}} \end{aligned} We see that there is one repeated variable factor, x. Therefore, the GCF of the expression is 16* x= 16x. Now we can write the given expression in terms of the GCF. 32x^2+48x ⇔ 16x* 2x+ 16x* 3 Finally, we will factor out the GCF. 16x* 2x+ 16x* 3 ⇔ 16x(2x+3)

We want to rewrite the given sum expression as an equivalent expression that is a product. For example, 20y^2-2y is a sum while 2y(10y-1) is a product. Let's look at our sum expression! - 40m-30 We can start by finding the greatest common factor (GCF) of the terms in the given expression. To do so, we will consider coefficients and variables separately. - 40m-30 ⇔ -( 40 m+ 30) Let's start by finding the GCF of 40 and 30 by listing all possible factors of both numbers. Factors of40:& 1,2,4,5,8, & 10, 20, and40 Factors of30:& 1,2,3,5,6, 10, & 15, and 30 We found that the GCF of 40 and 30 is 10. Since we factored out a negative sign, the GCF of the coefficients is - 10. To find the GCF of the variables, we need to identify the all possible variable factors in both terms. \begin{aligned} \textbf{Factors of }\bm{1^\textbf{st}}\textbf{ Variable:}&\ {\color{#FF0000}{m}}\\ \textbf{Factors of }\bm{2^\textbf{nd}}\textbf{ Variable:}&\ \end{aligned} We see that there are no repeated variable factors. Therefore, the GCF of the expression is - 10. Now we can write the given expression in terms of the GCF. - 40m-30 ⇔ - 10* 4m+( - 10)* 3 Finally, we will factor out the GCF. - 10* 4m+( - 10)* 3 ⇔ - 10(4m+3)

We want to rewrite the given sum expression as an equivalent expression that is a product. For example, - 15+30m is a sum while - 15(1+2m) is a product. Let's look at our sum expression! 63m^2-54m We can start by finding the greatest common factor (GCF) of the terms in the given expression. To do so, we will consider coefficients and variables separately. 63 m^2- 54 m Let's start by finding the GCF of 63 and 54 by listing all possible factors of both numbers. Factors of63:& 1,3,7, 9,21, and 63 Factors of54:& 1,2,3,6, 9,18, & 27, and 54 We found that the GCF of the coefficients is 9. To find the GCF of the variables, we need to identify the all possible variable factors in both terms. \begin{aligned} \textbf{Factors of }\bm{1^\textbf{st}}\textbf{ Variable:}&\ {\color{#FF0000}{m}}, m^2\\ \textbf{Factors of }\bm{2^\textbf{nd}}\textbf{ Variable:}&\ {\color{#FF0000}{m}} \end{aligned} We see that there is one repeated variable factor, m. Therefore, the GCF of the expression is 9* m= 9m. Now we can write the given expression in terms of the GCF. 63m^2-54m ⇔ 9m* 7m- 9m* 6 Finally, we will factor out the GCF. 9m* 7m- 9m* 6 ⇔ 9m(7m-6)