Question

a new company estimates its total profit (profit = total revenue minus total cost) as $p(x) = x^4 - 2x^3 - 240x - 35$, where $p$ is in hundreds of dollars and $x$ is the number of months elapsed since the companys start - up. according to the rational zero theorem, what can be the values of $x$ until the company breaks even?
a. 1, 3, 5, 7
b. 3, 5
c. 1, 5, 7, 35
d. 1, 35

Explanation:

Step1: Recall Rational Zero Theorem

The Rational Zero Theorem states that if a polynomial \( P(x)=a_nx^n + a_{n - 1}x^{n-1}+\dots+a_1x + a_0\) has integer coefficients, then every rational zero, \( \frac{p}{q}\), of \( P(x)\) has \( p\) as a factor of the constant term \( a_0\) and \( q\) as a factor of the leading coefficient \( a_n\).

For the polynomial \( P(x)=x^4 - 2x^3-240x - 35\), the leading coefficient \( a_n = 1\) (the coefficient of \( x^4\)) and the constant term \( a_0=- 35\).

Step2: Find factors of \( a_0\) and \( a_n\)

The factors of the leading coefficient \( a_n = 1\) are \( \pm1\). The factors of the constant term \( a_0=-35\) are \( \pm1,\pm5,\pm7,\pm35\).

By the Rational Zero Theorem, the possible rational zeros \( \frac{p}{q}\) are found by taking \( \frac{\text{factor of }a_0}{\text{factor of }a_n}\). Since the factor of \( a_n = 1\) is \( \pm1\), the possible rational zeros are \( \pm1,\pm5,\pm7,\pm35\). But since \( x\) represents the number of months elapsed, \( x>0\), so we consider only positive values: \( 1,5,7,35\).

Answer:

C. 1, 5, 7, 35