2022 AMC 10A Problem 16

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?

\dfrac{24}{5}

\dfrac{42}{5}

\dfrac{81}{5}

30

48

Answer: D

Concepts:Vieta’s Formulas polynomial volume

Difficulty rating: 1420

Small hint:

Use Vieta formulas for the three original dimensions

Big hint:

Expand the new volume before substituting symmetric sums

Solution:

Let

h, l,

and

w

be the dimensions of the old box. Then the volume of the new box is

(h + 2)(l + 2)(w + 2).

Expanding, we get

hlw + 2(hl + hw + lw)

+ 4(l + h + w) + 8.

We can use Vieta’s formulas to find the terms in this expression. We get that

hlw = -\dfrac{D}{A} = \dfrac{3}{5},

hl + hw + lw = \dfrac{C}{A} = \dfrac{29}{10},

and

l + h + w = -\dfrac{B}{A} = \dfrac{39}{10}.

Plugging these values into the expression, we get

\dfrac{3}{5} + 2 \cdot \dfrac{29}{10} + 4 \cdot \dfrac{39}{10} + 8 = 30.

Thus, D is the correct answer.