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Chapter Test

Determine the leading term, the leading coefficient, and the degree of the polynomial. Then classify the polynomial as constant, linear, quadratic, cubic, or quartic.

1. $f (x) = 2 x^{3} + 6 x^{2} - x^{4} + 11$ $f (x) = 2 x^{3} + 6 x^{2} - x^{4} + 11$
2. $h (x) = - 4.7 x + 29$ $h (x) = - 4.7 x + 29$
3. Find the zeros of the polynomial function and state the multiplicity of each:

$f (x) = x (3 x - 5) {(x - 3)}^{2} {(x + 1)}^{3} .$ $f (x) = x (3 x - 5) {(x - 3)}^{2} {(x + 1)}^{3} .$
4. Hybrid Automobiles. In 2004, only 84,199 hybrid automobiles were sold, while in 2012, 431,798 were sold (Source: U.S. Department of Transportation). The quartic function

$\begin{array}{rcl} f (x) & = & 897.690 x^{4} - 10,349.487 x^{3} \\ + 19,202.137 x^{2} + 91,597.838 x \\ + 88,209.580, \end{array}$ $\begin{array}{rcl} f (x) & = & 897.690 x^{4} - 10,349.487 x^{3} \\ + 19,202.137 x^{2} + 91,597.838 x \\ + 88,209.580, \end{array}$

where x is the number of years after 2004, can be used to estimate the number of hybrid automobiles sold in years 2004 to 2012. Use this function to estimate the number of hybrid automobiles sold in 2008 and in 2011.

Sketch the graph of the polynomial function.

5. $f (x) = x^{3} - 5 x^{2} + 2 x + 8$ $f (x) = x^{3} - 5 x^{2} + 2 x + 8$
6. $f (x) = - 2 x^{4} + x^{3} + 11 x^{2} - 4 x - 12$ $f (x) = - 2 x^{4} + x^{3} + 11 x^{2} - 4 x - 12$

Using the intermediate value theorem, determine, if possible, whether the function has a zero between a and b.

7. $f (x) = - 5 x^{2} + 3;$ $f (x) = - 5 x^{2} + 3;$ $a = 0, b = 2$ $a = 0, b = 2$
8. $g (x) = 2 x^{3} + 6 x^{2} - 3;$ $g (x) = 2 x^{3} + 6 x^{2} - 3;$ $a = - 2, b = - 1$ $a = - 2, b = - 1$
9. Use long division to find the quotient $Q (x)$ $Q (x)$ and the remainder $R (x)$ $R (x)$ when $P (x)$ $P (x)$ is divided by $d (x)$ $d (x)$ . Express $P (x)$ $P (x)$ in the form $d (x) \cdot Q (x) + R (x)$ $d (x) \cdot Q (x) + R (x)$ . Show your work.

$\begin{array}{rcl} P (x) & = & x^{4} + 3 x^{3} + 2 x - 5, \\ d (x) & = & x - 1 \end{array}$ $\begin{array}{rcl} P (x) & = & x^{4} + 3 x^{3} + 2 x - 5, \\ d (x) & = & x - 1 \end{array}$
10. Use synthetic division to find the quotient and the remainder. Show your work.

$(3 x^{3} - 12 x + 7) \div (x - 5)$ $(3 x^{3} - 12 x + 7) \div (x - 5)$
11. Use synthetic division to find $P (- 3)$ $P (- 3)$ for $P (x) = 2 x^{3} - 6 x^{2} + x - 4$ $P (x) = 2 x^{3} - 6 x^{2} + x - 4$ . Show your work.
12. Use synthetic division to determine whether −2 is a zero of $f (x) = x^{3} + 4 x^{2} + x - 6$ $f (x) = x^{3} + 4 x^{2} + x - 6$ . Answer yes or no. Show your work.
13. Find a polynomial function of degree 4 with −3 as a zero of multiplicity 2 and 0 and 6 as zeros of multiplicity 1.
14. Suppose that a polynomial function of degree 5 with rational coefficients has 1, $\sqrt{3}$ $\sqrt{3}$ , and $2 - i$ $2 - i$ as zeros. Find the other zeros.

Find a polynomial function of lowest degree with rational coefficients and the following as some of its zeros.

15. $- 10, 3 i$ $- 10, 3 i$
16. $0, - \sqrt{3}, 1 - i$ $0, - \sqrt{3}, 1 - i$

List all possible rational zeros.

17. $f (x) = 2 x^{3} + x^{2} - 2 x + 12$ $f (x) = 2 x^{3} + x^{2} - 2 x + 12$
18. $h (x) = 10 x^{4} - x^{3} + 2 x - 5$ $h (x) = 10 x^{4} - x^{3} + 2 x - 5$

For each polynomial function, (a) find the rational zeros and then the other zeros; that is, solve $f (x) = 0;$ $f (x) = 0;$ and (b) factor $f (x)$ $f (x)$ into linear factors.

19. $f (x) = x^{3} + x^{2} - 5 x - 5$ $f (x) = x^{3} + x^{2} - 5 x - 5$
20. $f (x) = 2 x^{4} - 11 x^{3} + 16 x^{2} - x - 6$ $f (x) = 2 x^{4} - 11 x^{3} + 16 x^{2} - x - 6$
21. $f (x) = x^{3} + 4 x^{2} + 4 x + 16$ $f (x) = x^{3} + 4 x^{2} + 4 x + 16$
22. $f (x) = 3 x^{4} - 11 x^{3} + 15 x^{2} - 9 x + 2$ $f (x) = 3 x^{4} - 11 x^{3} + 15 x^{2} - 9 x + 2$
23. What does Descartes’ rule of signs tell you about the number of positive real zeros and the number of negative real zeros of the following function?

$g (x) = - x^{8} + 2 x^{6} - 4 x^{3} - 1$ $g (x) = - x^{8} + 2 x^{6} - 4 x^{3} - 1$

Graph the function. Be sure to label all the asymptotes. List the domain and the x- and y-intercepts.

24. $f (x) = \frac{2}{{(x - 3)}^{2}}$ $f (x) = \frac{2}{{(x - 3)}^{2}}$
25. $f (x) = \frac{x + 3}{x^{2} - 3 x - 4}$ $f (x) = \frac{x + 3}{x^{2} - 3 x - 4}$
26. Find a rational function that has vertical asymptotes $x = - 1$ $x = - 1$ and $x = 2$ $x = 2$ and x-intercept (−4, 0).

Solve.

27. $2 x^{2} > 5 x + 3$ $2 x^{2} > 5 x + 3$
28. $\frac{x + 1}{x - 4} \leq 3$ $\frac{x + 1}{x - 4} \leq 3$
29. The function $S (t) = - 16 t^{2} + 64 t + 192$ $S (t) = - 16 t^{2} + 64 t + 192$ gives the height S, in feet, of a model rocket launched with a velocity of $64 ft / sec$ $64 ft / sec$ from a hill that is 192 ft high.
1. Determine how long it will take the rocket to reach the ground.
2. Find the interval on which the height of the rocket is greater than 240 ft.
30. The graph of $f (x) = x^{3} - x^{2} - 2$ $f (x) = x^{3} - x^{2} - 2$ is which of the following?

Synthesis

31. Find the domain of $f (x) = \sqrt{x^{2} + x - 12}$ $f (x) = \sqrt{x^{2} + x - 12}$ .

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Table of Contents for Chapter Test

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Table of Contents for
Chapter Test