Praxis Core Academic Skills for Educators: Mathematics (5733)Algebra and FunctionsHard
A company's production cost C (in thousands of dollars) is related to the number of units produced x by the equation C(x) = x^3 - 10x^2 + 31x - 30. If the cost is $0, which of the following could be the number of units produced?
- A6 units
- B2 units
- C8 units
- D4 units
Show answer & explanationAnswer & explanation
Correct answer: B. 2 units
To find the number of units produced when the cost is $0, we need to find the roots of the polynomial C(x) = x^3 - 10x^2 + 31x - 30 = 0. We can test the given options by substituting them into the equation.
Why the other options are wrong
- A. If x = 6, C(6) = (6)^3 - 10(6)^2 + 31(6) - 30 = 216 - 10(36) + 186 - 30 = 216 - 360 + 186 - 30 = 402 - 390 = 12. This is not 0.
- C. If x = 8, C(8) = (8)^3 - 10(8)^2 + 31(8) - 30 = 512 - 10(64) + 248 - 30 = 512 - 640 + 248 - 30 = 760 - 670 = 90. This is not 0.
- D. If x = 4, C(4) = (4)^3 - 10(4)^2 + 31(4) - 30 = 64 - 10(16) + 124 - 30 = 64 - 160 + 124 - 30 = 188 - 190 = -2. This is not 0.
Evaluating Polynomials
Evaluating a polynomial involves substituting a specific numerical value for the variable and then performing the indicated arithmetic operations to find the polynomial's value.
- Substitute the value for every instance of the variable.
- Follow the order of operations (PEMDAS/BODMAS).
- The result is the value of the polynomial at that specific point.
Memory trick: Test Each Option, Plug and Play.