Question 1

Given the function $$F(x) = x^4$ - 5x^3$ + 27x^2$ - 101x + 70$$, and knowing that $$x = 5 is a $$root, use synthetic division to find the resulting cubic polynomial after dividing $$F(x) by$$ $$(x-5). $$What is the resulting cubic polynomial?

Accepted Answer

$$x^3$$ + $$2x^2 + 17x - 14$

Question 2

The derivative of a power function $$x^n is $$given by which of the following formulas?

Accepted Answer

$$n x$$^{n-1}$$

Question 3

Given the cost function $$C(x) = 20x^3$ - 360x^2$ + 3200x - 1000$$, what is the formula for the average cost function $$A(x)$$?

Accepted Answer

$$A(x) = \frac{C(x)}{x}$$

Question 4

If $$F(x) = x^4$ - 5x^3$ + 27x^2$ - 101x + 70$$, and you divide all coefficients by their greatest common divisor (GCD), what is the resulting function $$F_1(x$$)$$?

Accepted Answer

$$F_1(x$$) = $$x^4$$ - $$5x^3$$ + $$27x^2 - 101x + 70$

Question 5

Newton's Method is used to approximate roots of a function $$F(x). $$Which of the following is the correct iterative formula for Newton's Method?

Accepted Answer

$$x_{n+1} = x_n - \frac{F(x_n)}{F'(x_n)}$$

Question 6

Suppose you have found a root $$r of$$ $$F(x)$$, and you divide $$F(x) by$$ $$(x - r)$$ using synthetic division, resulting in a cubic $$G(x). $$What is the degree of $$G(x)$$?

Accepted Answer

3

Question 7

Given $$G(x) = x^3$ + 2x^2$ + 17x - 14$$, use the quadratic formula to solve for the roots after factoring out one root. Which formula should you use?

Accepted Answer

$$x = \frac{-b \pm \sqrt{b^2$ - 4ac}}{2a}$$

Question 8

If a polynomial has four real roots, but only positive roots are considered for a production problem, which of the following best describes the correct approach?

Accepted Answer

Select the smallest positive root as the minimum of the cost function.

Question 9

Given the cost function $$C(x) = 20x^3$ - 360x^2$ + 3200x - 1000$$, find the critical points by setting the derivative equal to zero. What is the derivative $$C'(x)$$?

Accepted Answer

$$C'(x) = 60x^2$ - 720x + 3200$$

Question 10

A function $$F(x)$$ has a critical point at $$x = a if $$which of the following is true?

Accepted Answer

$$F'(a) = 0$$

Question 11

If $$F(x) is a $$quartic polynomial and you use synthetic division to remove a root $$r$$, what is the degree of the resulting polynomial?

Accepted Answer

3

Question 12

Which of the following best describes the purpose of finding the critical points of a function in a real-world context?

Accepted Answer

To determine where the function has local maxima or minima, such as minimum cost or maximum profit.

Question 13

If you have a cubic polynomial $$G(x)$$ and you use the quadratic formula to find its roots, what must you do first?

Accepted Answer

Factor out one root to reduce it to a quadratic.

Question 14

Suppose you are using Newton's Method to find a root of $$F(x)$$ and your initial guess is far from the actual root. What is a likely outcome?

Accepted Answer

The method may not converge to the correct root.

Question 15

If the derivative of a function at a point is zero, what does this indicate about the function at that point?

Accepted Answer

It may be a local maximum, minimum, or a saddle point.

Question 16

Given $$F(x) = x^4$ - 5x^3$ + 27x^2$ - 101x + 70$$, and you have found all real roots, which root should you select to minimize a cost function if only positive values are allowed?

Accepted Answer

The smallest positive root.

Roots, Derivatives, and Applications in College Algebra

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