Role Consider yourself as an employee of that well-known restaurant Audier ager of the restaurant Situation The supply of pako to that well-known restaurant is represented by equation \( \mathrm{P}(\mathrm{x})=\mathrm{x} 3-9 \times 2+23 \mathrm{x}-15 \), (where x is the number of months) the employee, after how many months to wait such that the supply bec Product/ Performance and Purpose Answer the following guide questions: 1. In what months (ordinal number) the "pako" supply becomes zero? 2. How many times the supply becomes zero? 3. What is the equation of the polynomial in factored form? 4. How many months to wait such that the supply becomes stable?

To analyze the supply of "pako" represented by the polynomial equation \( P(x) = x^3 - 9x^2 + 23x - 15 \), we will address each of the guide questions step by step. ### 1. In what months (ordinal number) does the "pako" supply become zero? To find the months when the supply becomes zero, we need to solve the equation \( P(x) = 0 \): \[ x^3 - 9x^2 + 23x - 15 = 0 \] We can use the Rational Root Theorem to test possible rational roots. The possible rational roots are the factors of the constant term (-15) divided by the factors of the leading coefficient (1). The possible rational roots are \( \pm 1, \pm 3, \pm 5, \pm 15 \). Testing these values: - For \( x = 1 \): \[ P(1) = 1^3 - 9(1^2) + 23(1) - 15 = 1 - 9 + 23 - 15 = 0 \] So, \( x = 1 \) is a root. Now we can perform synthetic division of \( P(x) \) by \( (x - 1) \): \[ \begin{array}{r|rrrr} 1 & 1 & -9 & 23 & -15 \\ & & 1 & -8 & 15 \\ \hline & 1 & -8 & 15 & 0 \\ \end{array} \] This gives us the quotient \( x^2 - 8x + 15 \). Now we can factor this quadratic: \[ x^2 - 8x + 15 = (x - 3)(x - 5) \] Thus, the complete factorization of \( P(x) \) is: \[ P(x) = (x - 1)(x - 3)(x - 5) \] Setting each factor to zero gives us the months when the supply becomes zero: - \( x - 1 = 0 \) → \( x = 1 \) - \( x - 3 = 0 \) → \( x = 3 \) - \( x - 5 = 0 \) → \( x = 5 \) **Answer:** The "pako" supply becomes zero in months 1, 3, and 5. ### 2. How many times does the supply become zero? From the factorization, we see that the supply becomes zero at three distinct months (1, 3, and 5). **Answer:** The supply becomes zero 3 times. ### 3. What is the equation of the polynomial in factored form? As derived above, the polynomial in factored form is: \[ P(x) = (x - 1)(x - 3)(x - 5) \] **Answer:** The equation of the polynomial in factored form is \( P(x) = (x - 1)(x - 3)(x - 5) \). ### 4. How many months to wait such that the supply becomes stable? The supply becomes stable when it is no longer zero, which occurs after the last zero crossing. The last month when the supply is zero is month 5. Therefore, the supply will be stable starting from month 6. **Answer:** The supply becomes stable after 5 months, so you need to wait 6 months.