Question

1. the volume of a rectangular prism is modeled with the function $v(x) = 2x^3 + x^2 - 13x + 6$. the prism has a height of $(x - 2)$. find one possible solution set for the length and width of the prism. ($v = lwh$)

a) $(2x - 1)(x + 3)$ b) $(2x + 1)(x - 3)$ c) $(x - 1)(x + 3)$ d) $(x - 2)(x + 3)$

options: c, b, d, a

Explanation:

Step1: Recall Volume Formula

The volume of a rectangular prism is \( V = LWH \). We know \( V(x) = 2x^3 + x^2 - 13x + 6 \) and height \( H=(x - 2) \). So we need to factor \( V(x) \) by dividing by \( (x - 2) \) (using polynomial long division or synthetic division) and then factor the quotient.

Step2: Synthetic Division for \( V(x)\div(x - 2) \)

The coefficients of \( V(x) \) are \( 2, 1, -13, 6 \). Using synthetic division with root \( 2 \):

$$
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$$

So the quotient is \( 2x^2 + 5x - 3 \).

Step3: Factor the Quotient

Factor \( 2x^2 + 5x - 3 \). We need two numbers that multiply to \( 2\times(-3)= -6 \) and add to \( 5 \). The numbers are \( 6 \) and \( -1 \).
Rewrite the middle term: \( 2x^2 + 6x - x - 3 \)
Group: \( (2x^2 + 6x) + (-x - 3) = 2x(x + 3) - 1(x + 3) = (2x - 1)(x + 3) \)

Answer:

a) \( (2x - 1)(x + 3) \)