SAT HELP. I'm not sure if I did this right so can someone explain in detail and give the answer so I can cross check.

Question

A manufacturing company wants to keep their revenue positive. The equation for C(x) represents their cost, where (x) represents the time in months. The equation for P(x) represents their profit. The equation for R(x) represents their revenue.

C(x) = 3x^(3) + 450x^(2)

R(x) = 7x^(3) + 300x^(2) + 900x

P(x) = R(x) - C(x)

a. Write an equation P(x) to represent the profit.

b. Identify the degree, leading coefficient, leading term, and constant of the profit equation.

c. Factor the polynomial.

d. Solve the equation to determine the values where the company will break even.

Answer 1

P(x) = R(x) - C(x)

P(x) = (7x³ + 300x² + 900x) - (3x³ + 450x²)

P(x) = 7x³ + 300x² + 900x - 3x³ - 450x²

P(x) = 4x³ - 150x² + 900x

P(x) has a degree of 3, a leading coefficient of 4, a leading term of 4x³ and a constant of zero (0)

P(x) in factored form:

P(x) = 2x(2x² - 75x + 450)

P(x) = 2x(2x - 15)(x - 30)

The company breaks even when profit = zero so:

0 = 2x(2x - 15)(x - 30)

to solve, set each piece equal to zero:

2x = 0 when x = 0

2x - 15 = 0 when x = 7.5

x - 30 = 0 when x = 30

Meaning there will be zero profit (or a "break even" event) at zero months, 7.5 months, and 30 months.