The dimensions of a rectangular box are given by the expressions 5x, (2x - 3), and (x + 4)… — Armed Services Vocational Aptitude Battery (ASVAB)

The dimensions of a rectangular box are given by the expressions $5x$ , $(2x - 3)$ , and $(x + 4)$ . The volume of the box is the product of its three dimensions. When the volume is expressed as a fully simplified polynomial in the form $ax^3 + bx^2 + cx$ , determine the value of the coefficient $b$ .

Answer:The value of

b

is 【25】.

Answer

To find the coefficient

b

, you must expand the product of the three dimensions. First, multiply the binomials using FOIL:

(2x - 3)(x + 4) = 2x^2 + 8x - 3x - 12 = 2x^2 + 5x - 12

. Then, distribute

5x

across the resulting trinomial:

5x(2x^2 + 5x - 12) = 10x^3 + 25x^2 - 60x

. Comparing this to the specified polynomial form

ax^3 + bx^2 + cx

, the term with

x^2

25x^2

, making the coefficient

b = 25

Step-by-Step Solution

Set up the volume expression by multiplying the three given dimensions.

V = 5x(2x - 3)(x + 4)

The volume of a rectangular box is calculated as length

\times

width

\times

height.

Multiply the two binomials using the FOIL method.

(2x - 3)(x + 4) = 2x^2 + 8x - 3x - 12 = 2x^2 + 5x - 12

It is mathematically sound and often easier to multiply the binomials first before distributing the monomial.

Distribute the monomial

5x

to each term in the resulting trinomial.

5x(2x^2 + 5x - 12) = 10x^3 + 25x^2 - 60x

To fully expand the polynomial, every term inside the parentheses must be multiplied by

5x

Identify the value of

b

by matching the result to the given standard form

ax^3 + bx^2 + cx

b = 25

The variable

b

represents the coefficient of the

x^2

term in the fully expanded polynomial.

Key Concept

Multiplying Polynomials and the Distributive Property

Question

Answer

Step-by-Step Solution

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