Exercise 40 Page 532

Recall that the volume of a prism is obtained by multiplying one of the parallel polygons and the prism's height. In this case, the base is a rectangle. Since the area of a rectangle is obtained by multiplying its length times its width, we can write the formula for the volume of this prism as shown below. V &= Area of one parallel side * Height V &= Length * Width * Height In accordance to the volume formula, to find expressions representing the dimensions of the pedestal we can factor the polynomial representing its volume in three factors. Let's start by finding the greatest common factor (GCF) between all the terms. To do this, we can write the terms of the given polynomial, 63x^3-28x, as a product of their prime factors to find the common ones. 63x^3& = 3 * 3 * 7 * x * x * x 28x & = 2 * 2 * 7 * 11 * x

Therefore, the GCF is 7 * x = 7x. We can now simplify our polynomial by factoring 7x out. 63x^3-28x ⇕ 7x(9x^2-4) Notice that the binomial inside the parentheses is a difference of squares. Therefore, we can factor it as the product of the sum and difference of two terms by using the rule shown below. a^2-b^2 = (a+b)(a-b) Let's give it a try. a^2-b^2 = (a+b)(a-b) [1em] 9x^2-4 = (3x+2)(3x-2) With this, we can write the exercise's polynomial in its completely factored form. a^2-b^2 = (a+b)(a-b) ⇕ 7x(3x+2)(3x-2) In accordance to the volume formula, we can identify each factor as a dimension of the pedestal.