2022 AMC 10 A

Complete problem set with solutions and individual problem pages

Problem 16 Easy

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?

  • A.

    \frac{24}{5}

  • B.

    \dfrac{42}{5}

  • C.

    \frac{81}{5}

  • D.

    30

  • E.

    48

Answer:D

Assume the roots of the polynomial are a,b,c.

According to Vieta's formula,

\begin{cases}a+b+c=\frac{39}{10} \\ ab+bc+ca=\frac{29}{10} \\ abc=\frac 35 \end{cases}

(a+2)(b+2)(c+2)=abc+2(ab+bc+ca)+4(a+b+c)+8=30.

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