Q: For the polynomial $$f(x) = (x - 2)^2$(x + 1)^3$, what is the end behavior as x$$ \to \infty$$?

As $$x \to \infty$$, $$f(x) \to \infty$$

Q: For $$f(x) = x^4$ - 5x^2$ + 4$$, what is the end behavior as $$x \to -\infty$$?

As $$x \to -\infty$$, $$f(x) \to \infty$$

Q: For $$f(x) = - (x + 3)^2$ (x - 1) (2x - 4)^3$, what is the end behavior as x$$ \to -\infty$$?

As $$x \to -\infty$$, $$f(x) \to -\infty$$

Question 1

A ball is thrown upward from the ground. Its height after $$t$$ seconds is given by $$h(t) = -16t^2$ + 64t + 3. At $$what time does the ball reach its maximum height?

Accepted Answer

$$t = 4$$ seconds

Question 2

Given the profit function $$P(x) = -2x^2$ + 80x - 300$$, what is the maximum profit achievable?

Accepted Answer

$$700$$ dollars

Question 3

For the profit function $$P(x) = -2x^2$ + 80x - 300$$, how many items must be sold to achieve the maximum profit?

Accepted Answer

$$40$$

Question 4

You have 200 yards of fencing to build a rectangular enclosure. What is the maximum area you can enclose?

Accepted Answer

$$10,000$$ \, 	ext{yd}^2$

Question 5

For the fencing problem, what are the dimensions of the rectangle that maximize the area?

Accepted Answer

$$100 	imes 100$$ yards

Question 6

A skateboarder’s vertical height is given by $$H(x) = 4x^2$ - 24x + 20. At $$what value(s) of $$x is $$the skateboarder at the top of the ramp (height $$= 0$$)?

Accepted Answer

$$x = 1$$ and $$x = 5$$

Question 7

For $$H(x) = 4x^2$ - 24x + 20$$, at what value of $$x is $$the skateboarder at the bottom of the ramp?

Accepted Answer

$$x = 3$$

Question 8

For the polynomial $$f(x) = (x - 2)^2$(x + 1)^3$, what is the end behavior as x$$ 	o \infty$$?

Accepted Answer

As $$x 	o \infty$$, $$f(x) 	o \infty$$

Question 9

For $$f(x) = - (x + 3)^2$ (x - 1) (2x - 4)^3$, what is the degree of the polynomial?

Accepted Answer

Degree $$7$$

Question 10

For $$f(x) = 2(x^2$ - 25)(x - 5)^4$, how many distinct real zeros does the polynomial have?

Accepted Answer

3

Question 11

For $$f(x) = - (x + 4)(x - 2)^2$(x + 1)^2$, what is the multiplicity of the zero at x = -1$?

Accepted Answer

2

Question 12

Solve $$f(x) = x^3$ - 4x^2$ - x + 4 = 0. $$Which of the following is a real solution?

Accepted Answer

$$x = 4$$

Question 13

For $$f(x) = x^4$ - 5x^2$ + 4$$, what are the real zeros?

Accepted Answer

$$x = 2, x = -2, x = 1, x = -1$$

Question 14

For $$f(x) = x^4$ - 5x^2$ + 4$$, what is the end behavior as $$x 	o -\infty$$?

Accepted Answer

As $$x 	o -\infty$$, $$f(x) 	o \infty$$

Question 15

A chemical reaction produces a profit modeled by $$P(x) = -2x^2$ + 80x - 300$$, where $$x is $$the number of batches produced. If the company wants to maximize profit, how many batches should they produce?

Accepted Answer

$$40$$

Question 16

A rectangular garden is to be fenced with 200 yards of fencing. If the area is maximized, what is the shape of the garden?

Accepted Answer

A square

Question 17

For $$f(x) = (x - 2)^2$(x + 1)^3$, what is the behavior of the graph at x = -1$?

Accepted Answer

The graph crosses the $$x-$$axis and flattens at $$x = -1$$

Question 18

For $$f(x) = - (x + 3)^2$ (x - 1) (2x - 4)^3$, what is the end behavior as x$$ 	o -\infty$$?

Accepted Answer

As $$x 	o -\infty$$, $$f(x) 	o -\infty$$

Question 19

For $$f(x) = 2(x^2$ - 25)(x - 5)^4$, what is the multiplicity of the zero at x = 5$?

Accepted Answer

4

College Algebra Optimization and Polynomial Analysis Guidance

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