Algebra 2 help

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

C(x) = 3x³ + 450x²

R(x) = 7x³ + 300x² + 900x

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

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

= 7x³ + 300x² + 900x - 3x³ - 450x²

= 4x³ - 150x² + 900x

b.) The degree is 3 because that's the highest exponent of the polynomial (profit equation). The leading coefficient is 4. The leading term is 4x³. The constant is zero.

c.) Factor using GCF (greatest common factor).

4x³ - 150x² + 900x = x(4x² - 150x + 900) = 2x(2x² - 75x + 450)

d.) Lets apply the zero product property. Set 2x and 2x² - 75x + 450 equal to zero.

2x = 0 ⇒ x = 0 or 2x² - 75x + 450 = 0

Use the quadratic formula to solve for x with a = 2, b = -75, and c = 450.

x = (-b ± √b² - 4ac) / 2a = [-(-75) ± √75² - 4(2)(450)] / 2(2)

= (75 ± √5625 - 3600) / 4

= (75 ± √2025) / 4

= (75 ± 45) / 4

x = (75 + 45) / 4 = 120 / 4 = 30 or x = (75 - 45) / 4 = 30 / 4 = 7.5

The solutions are x = 0, 7.5, and 30. The company will break even at the initial start (0 months), 7.5 months, and 30 months (2.5 years).