smc

Question

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?

Accepted Answer

Let a, b, c be the three roots of the polynomial. The lengthened prism's volume 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. By Vieta's formulas, we know that a cubic polynomial Ax^3+Bx^2+Cx+D with roots a, b, c satisfies: alignat*8 a+b+c &= -BA &&= 3910, ab+ac+bc &= 2mmCA &&= 2910, abc &= -DA &&= 610. alignat* We can substitute these into the expression, obtaining V = 610 + 2(2910) + 4(3910) + 8 = 30.