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 rectangle box is formed by lengthening each edge by 2 units.

Question

What is the volume of the new box?

Answer 1

Using the rational roots theorem the possible roots are:

(±) 1/5, 2/5, 1/2, 3/5, 1, 6/5,3/2, 2, 3, 6

Using these possibilities you will find one of three rational roots by substituting the right root where the equation will equal zero.

One such root is 3.

If x = 3, it follows that x - 3 = 0 and thus (x - 3) is a factor.

Using either polynomial long division or synthetic division and dividing (x - 3) into the original equation yields:

(10x³ - 39x² + 29x - 6)/ (x - 3) = (10x² - 9x + 2)

At this point you can use the RRT again, the quadratic formula or factoring:

Using factoring we get:

10x² - 9x + 2 = (5x - 2)(2x - 1)

So:

5x - 2 = 0; x = 2/5 (0.4), or:

2x - 1 = 0; x = 1/2 (0.5)

Roots are therefore:

x = 0.4, 0.5, 3

If these correspond to the height, length and width, the new height, length and width are:

H = 0.4 + 2 = 2.4; L = 0.5 + 2 = 2.5; W = 3 + 2 = 5, and the volume is:

V = 2.4 * 2.5 * 5 = 30 sq. units