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

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

Q: For $$f(x) = x^3$ - 4x^2$ - x + 4$$, 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 $$x \to -\infty$$, $$f(x) \to \infty$$

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

$$800$$

Question 3

If you have 200 yards of fencing to build a rectangular enclosure, what are the dimensions that maximize the area?

Accepted Answer

$$50$$ yards by $$50$$ yards

Question 4

A skateboarder’s vertical height on a ramp is given by $$H(x) = 4x^2$ - 24x + 20. At $$what value of $$x$$ does he reach the bottom of the ramp?

Accepted Answer

$$x = 3$$

Question 5

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 6

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

Accepted Answer

Multiplicity $$2$$

Question 7

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

Accepted Answer

3

Question 8

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

Accepted Answer

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

Question 9

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

Accepted Answer

$$x = 4$$

Question 10

For $$f(x) = x^4$ - 5x^2$ + 4$$, what is the total number of real roots?

Accepted Answer

4

Question 11

A company models its profit with $$P(x) = -2x^2$ + 80x - 300. $$How many items must be sold to achieve maximum profit?

Accepted Answer

$$x = 20$$

Question 12

If a rectangular fence is to be built with 200 yards of fencing, what is the maximum area that can be enclosed?

Accepted Answer

$$2,500$$ square yards

Question 13

For the polynomial $$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 due to multiplicity 3

Question 14

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

Accepted Answer

Degree $$6$$

Question 15

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

Accepted Answer

Multiplicity $$5$$

Question 16

For $$f(x) = x^4$ - 5x^2$ + 4$$, which of the following is NOT a root?

Accepted Answer

$$x = 3$$

Question 17

A ball is thrown upward with height $$h(t) = -16t^2$ + 64t + 3. $$What is the maximum height reached by the ball?

Accepted Answer

$$131$$ feet

Question 18

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

Accepted Answer

The graph touches the $$x-$$axis and turns around due to multiplicity 2

Question 19

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

Accepted Answer

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

Question 20

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

Accepted Answer

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

College Algebra Optimization and Polynomial Analysis Guidance

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