Q: A chemical reaction rate is modeled by $$R(t) = 5e$$^{-0.2t}$$. What is the initial rate and what happens to the rate as t$$ \to \infty$$?

Initial rate is 5, rate approaches 0 as $$t \to \infty$$

Question 1

Solve for $$x in $$the equation $$g(6x^3) - 7 = 9$ if g(x) = 2x - 5$. What is the value of x$?

Accepted Answer

$$x = 1.5$$

Question 2

Divide the polynomial $$4x^3$$ - $$5x^2 + 7x - 5$ by x^2$ + 4x^2$ using long division. What is the quotient?

Accepted Answer

$$4 - 5x + 7x - 5$$

Question 3

Given the function $$f(x) = -6(x+3)(x-2)^2$(x+1)^2(x+4)^3$, which of the following describes the end behavior of f(x$$)$$ as x$$ 	o \infty$$?

Accepted Answer

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

Question 4

Divide $$x^3$$ + $$12x^2 - 4x + 10$ by x + 4$ using synthetic division. What is the remainder?

Accepted Answer

$$-6$$

Question 5

List all possible rational zeros for the polynomial $$f(x) = -6x^4$ - 2x^3$ + 5x + 13$$ using the Rational Root Theorem.

Accepted Answer

$$\pm1, \pm13, \pm2, \pm5, \pm6, \pm26$$

Question 6

Express $$\log_5$ \left( \sqrt{\dfrac{x^2$ + 5x - 21}{5x^2$ + 1}} \right) as a $$sum or difference of the simplest possible terms.

Accepted Answer

$$\dfrac{1}{2} \log_5$(x^2 + 5x - 21$$) - \dfrac{1}{2} \$$log_5(5x^2$ + 1)$$

Question 7

Find a third degree polynomial that has the given zeros: $$-1$$, $$-2$$, and $$3.$$

Accepted Answer

$$(x+1)(x+2)(x-3)$$

Question 8

Solve for $$x$$: $$2\log_2$(x) - \log_2$(4) = 4$$

Accepted Answer

$$x = 16$$

Question 9

Given $$f(x) = \dfrac{-2x^2$ + 12}{3x^2$ + 8x + 2}$$, what is the horizontal asymptote of the function?

Accepted Answer

$$y = 0$$

Question 10

Using Descartes' Rule of Signs, determine the possible number of positive real zeros for $$f(x) = 4x^7$ + x^6$ - 2x^5$ + 3x^4$ + 9x^2$ - 6x^2$ - 8x - 10.$$

Accepted Answer

4 or 2 or 0

Question 11

A chemical reaction rate is modeled by $$R(t) = 5e$$^{-0.2t}$$. What is the initial rate and what happens to the rate as t$$ 	o \infty$$?

Accepted Answer

Initial rate is 5, rate approaches 0 as $$t 	o \infty$$

Question 12

If $$f(x) = x^4$ - 2x^3$ + 5x^2$ - 8x + 10$$, which of the following is a possible rational zero?

Accepted Answer

$$x = 1$$

Question 13

A population grows according to $$P(t) = 1000e$$^{0.03t}$$. How long will it take for the population to double?

Accepted Answer

$$t = \dfrac{\ln 2}{0.03}$$

Question 14

Which of the following describes the multiplicity of the zero $$x = -2$$ for $$f(x) = (x+2)^3$(x-1)^2$?

Accepted Answer

Multiplicity 3

Question 15

Given $$f(x) = \dfrac{2x^2$ - 3x + 1}{x^2$ - 4}$$, what are the $$x-$$values where the function is undefined?

Accepted Answer

$$x = 2$$ and $$x = -2$$

Question 16

A medical dosage is modeled by $$D(t) = 50e$$^{-0.1t}$$. What is the dosage after 10 hours?

Accepted Answer

$$D(10) = 50e$$^{-1}$$

Question 17

Which of the following is the sum of $$\log_3$(9) + \log_3$(27)$$?

Accepted Answer

$$5$$

Question 18

If $$f(x) = x^3$ - 4x^2$ + 7x - 10$$, which of the following is a factor of $$f(x)$$?

Accepted Answer

$$x - 10$$

Question 19

A company’s profit is modeled by $$P(x) = -2x^2$ + 8x - 12. $$What is the maximum profit and at what value of $$x$$ does it occur?

Accepted Answer

Maximum profit is $$-4 at$$ $$x = 2$$

Precalculus Polynomial and Rational Functions: Study Notes and Practice Problems

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