The volume of a water tank, in cubic feet, is given by the polynomial V(t) = 3t(2t - 1)(t … — ACT

The volume of a water tank, in cubic feet, is given by the polynomial $V(t) = 3t(2t - 1)(t + 4)$ , where $t$ represents the time in hours since the pump was turned on. When the polynomial is written in standard form, what is the coefficient of the $t^2$ term?

Answer: 21

Answer

To find the coefficient of the

t^2

term, we perform polynomial multiplication on the expression

V(t) = 3t(2t - 1)(t + 4)

. First, multiplying the binomials

(2t - 1)(t + 4)

yields

2t^2 + 7t - 4

. Next, distributing the

3t

term to the trinomial yields

6t^3 + 21t^2 - 12t

. The coefficient of the

t^2

term in this standard form polynomial is 21.

Step-by-Step Solution

Expand the product of the two binomials

(2t - 1)(t + 4)

2t^2 + 7t - 4

Use the distributive property (FOIL) to multiply the binomials:

2t(t) + 2t(4) - 1(t) - 1(4) = 2t^2 + 8t - t - 4 = 2t^2 + 7t - 4

Multiply the resulting trinomial by the monomial

3t

6t^3 + 21t^2 - 12t

Distribute

3t

to each term:

3t(2t^2) + 3t(7t) + 3t(-4) = 6t^3 + 21t^2 - 12t

Identify the coefficient of the

t^2

term.

The coefficient is the numerical factor of the

t^2

term, which is 21.

Key Concept

Operations on Polynomials

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