Question

1. pete has a pressure washing business. he has a truck with a big tank of water, which is used throughout the day. the amount of water (in gallons) remaining in the tank after x hours is modeled by the function below. find all of the zeros of the function, and interpret them in context.

$y = x^3 - 7x^2 + 2x -14$

Explanation:

Step1: Group the terms

Group the first two terms and the last two terms:
$$(x^{3}-7x^{2})+(2x - 14)$$

Step2: Factor out the common factors from each group

Factor out \(x^{2}\) from the first group and \(2\) from the second group:
$$x^{2}(x - 7)+2(x - 7)$$

Step3: Factor out the common binomial factor

Factor out \((x - 7)\):
$$(x - 7)(x^{2}+2)$$

Step4: Set each factor equal to zero and solve for \(x\)

Set \(x-7=0\), then \(x = 7\).
Set \(x^{2}+2=0\), then \(x^{2}=-2\), and \(x=\pm\sqrt{-2}=\pm\sqrt{2}i\) (complex roots).

Since \(x\) represents hours (a real - world quantity, non - complex), we consider only the real root.

Answer:

The zero of the function is \(x = 7\). In the context of the problem, this means that after \(7\) hours, the amount of water in the tank is \(0\) gallons.