Lesson goal: Polynomial Roots and Box Volume (AMC 10A Problem 16)

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In the 2022 AMC 10A competition, Problem #16 connects polynomial roots, geometric volume, and Vieta's formulas:

Contest Problem Statement:
"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?"

Method 1: Factoring the Polynomial

Let the original dimensions be the three roots $r_1, r_2, r_3$. Using CodeByMath's symbolic factor() function:
$$10x^3 - 39x^2 + 29x - 6 = (x - 3)(2x - 1)(5x - 2)$$
Setting each factor to zero reveals the original edge lengths:
  • $x - 3 = 0 \implies r_1 = 3$
  • $2x - 1 = 0 \implies r_2 = \frac{1}{2} = 0.5$
  • $5x - 2 = 0 \implies r_3 = \frac{2}{5} = 0.4$
Lengthening each dimension by $2$ units gives:
  • New height: $3 + 2 = 5$
  • New length: $0.5 + 2 = 2.5$
  • New width: $0.4 + 2 = 2.4$
The volume of the new box is simply the product:
$$V_{\text{new}} = 5 \times 2.5 \times 2.4 = 30$$

Method 2: The Elegant Polynomial Shift Identity

Notice how the polynomial can be written in factored form:
$$P(x) = 10(x - r_1)(x - r_2)(x - r_3)$$
What happens if we evaluate $P(-2)$?
$$P(-2) = 10(-2 - r_1)(-2 - r_2)(-2 - r_3) = -10(r_1 + 2)(r_2 + 2)(r_3 + 2)$$
Notice that $(r_1 + 2)(r_2 + 2)(r_3 + 2)$ is precisely the volume of the new box $V_{\text{new}}$!
Therefore:
$$P(-2) = -10 \times V_{\text{new}} \implies V_{\text{new}} = -\frac{P(-2)}{10}$$
Evaluating $P(-2)$ directly:
$$P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -80 - 156 - 58 - 6 = -300$$
$$V_{\text{new}} = -\frac{-300}{10} = 30$$
We found the new volume without ever needing to compute the individual roots!
factor("10*x^3 - 39*x^2 + 29*x - 6")
r1 = 3; r2 = 1/2; r3 = 2/5
vol_new = (r1 + 2) * (r2 + 2) * (r3 + 2)
p_neg2 = 10 * (-2)^3 - 39 * (-2)^2 + 29 * (-2) - 6
vol_shift = -p_neg2 / 10
Move the mouse over a dotted box for more information.

Both methods yield exactly 30, matching option (D) 30 on the 2022 AMC 10A exam.

Now you try. Replace ???? with 2 in each factor and click Run to compute the new box volume (28.8). Then check Example 1 to see polynomial expansion! for the new box $(x - (r_1+2))(x - (r_2+2))(x - (r_3+2))$ looks!

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