2022 AMC 12A Problems/Problem 15

Problem

The roots of the polynomial

$10x^3 - 39x^2 + 29x - 6$

are the height, length, and width of a rectangular box (right rectangular prism. A new rectangular box is formed by lengthening each edge of the original box by 2 units. What is the volume of the new box?

Solution

Let $a$ , $b$ , $c$ be the three roots of the polynomial. The lenghtened prism's area is $V = (a+2)(b+2)(c+2) = abc+2ac+2ab+2bc+4a+4b+4c+8 = abc + 2(ab+ac+bc) + 4(a+b+c) + 8$ .