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Final Exam Review MATH 1021 Spring 2022

Lecture 04

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College Algebra (MATH 1021)

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Math 1021 Final Exam Review (2, 2, 4, 4, 5.1-5 inclusive, 9, 5, 5.

and 6.1-6 inclusive, 6, 7, 7, 7, 7, 8, 8, 8, and 8)

NOTE THIS IS NOT ALL ENCOMPASSING. THERE MIGHT BE TYPES OF

PROBLEMS ON THE TEST THAT ARE NOT ON THIS REVIEW. You must

know how to do any of the types of homework problems that were assigned. Any

problem similar to a sample problem or a homework problem may appear on the

test. You are also responsible for the examples worked out in each assigned section

in the textbook even though they are not done in class.

Please review problems from Test 1, Test 2 and Test 3 review packets as well.

1. Determine whether the correspondence is a function. Justify your answer.

2. 3. Find the domain. Express answers in interval notation.

(a) 𝑓(𝑥) = 𝑥ଶ − 2𝑥 + 1 (b) 𝑔(𝑥) =

####### ଻

####### ହି௫

(c ) 𝑇(𝑥) = |2𝑥 − 3|

(d) 𝐹(𝑥) =

####### ௫యି ௫ మା௫ାଶ

####### ௫మାଵଶ௫ାଷହ

(e ) 𝑄(𝑥) = √5 − 3𝑥 (f) 𝑔(𝑥) =

####### ଵ

####### √௫

4. 𝑰𝒇 𝒇(𝒙) = −𝟓𝒙 + 𝟐 𝒂𝒏𝒅 𝒈(𝒙) = 𝒙𝟐 . Find

(a) (𝑓 + 𝑔)(𝑥) (b) (𝑔 − 𝑓)(𝑥) (c) (𝑓 ∙ 𝑔)(−9) (d) (

####### ௙

####### ௚

)(𝑥)

(e) (𝑓𝑜𝑔)(𝑥) (f) (𝑔𝑜𝑓)(𝑥)

5. Multiply the following polynomials.

(a) (3𝑥ଷ𝑦ହ)(−4𝑥ସ𝑦଺) (b) (2𝑥ଷ − 3𝑥ଶ + 𝑥 + 5)(2𝑥ଶ + 𝑥 − 1)

(c ) (2𝑥 + 3𝑦)ଶ (d) (𝑥 + 2)(𝑥 − 2)(𝑥ଶ + 4)

(e) (2𝑥 + 3𝑦)(2𝑥 − 3𝑦) (f) −4𝑥ହ(2𝑥ଷ − 3𝑥ଶ + 𝑥 + 5)

(g) (2𝑥 − 3𝑦)

####### ଷ

6. Factor completely.

(a) 𝑥ଶ − 10𝑥 + 25 (b) 4𝑥ଶ + 12𝑥 + 9 (c) 9𝑥ଶ − 4

(d) 6𝑥ଶ + 10𝑥 − 3𝑥 − 5 (e) 𝑥ଶ + 4𝑥 − 12

(f) 2𝑥ସ − 16𝑥 (g) 2𝑥ଶ + 𝑥 − 3 (h) 12𝑥ଶ + 7𝑥 − 10

(i) 2𝑥𝑦ଷ − 22𝑥𝑦ଶ + 48𝑥𝑦 (j) 81𝑥ଶ − 36𝑥 + 4 (k) 54𝑥଺𝑦 − 16𝑦

7. Perform the indicated operations and simplify your answers.

(a)

####### ௫

####### ௫ିଷ

+

####### ଷ

####### ଷି௫

(b)

####### ଵ

####### ସ௫మ

−

####### ଶ௫ାଵ

####### ଷ௫య

+

####### ଷ

####### ଵଶ

(c)

####### ௬ିଷ

####### ௬మି ସ

−

####### ௬ାଶ

####### ௬మି ସ௬ାସ

−

####### ଶ

####### ଶି௬

(d)

####### ௬మି ଵ଴௬ାଽ

####### ௬మ ିଵ

∙

####### ௬ାସ

####### ௬మି ହ௬ିଷ଺

(e)

####### ௫ାଵ

####### ௫ି௫ మ

⋅

####### ௫మି ଶ௫ାଵ

####### ௫మ ିଵ

(f)

####### ସ௫మି ସ௫ାଵ

####### ଶ௫మାହ௫ିଷ

÷

####### ଶ௫మ ିଷ௫ିଶ

####### ଶ௫మା଻௫ାଷ

(g)

####### ସ௫మି ଽ௬ మ

####### ଼௫ యି ଶ଻௬ య

÷

####### ସ௫మାଵଶ௫௬ାଽ௬మ

####### ସ௫మା଺௫௬ାଽ௬మ

16. Express the following in terms of rational exponents.

(a)  

5

3 1 



 



 x  (b) ቀඥ(𝑥 − 1)(𝑥 − 2)ቁ

ଷ

(c ) ඥ√𝑥

య

17. Simplify the following, Express in terms of positive exponents. Variables

represent positive values.

(a)  

323 27 x (b)  16 x 8 y 4  14 (c)

6 14 13

13 12















x y

(d) 3 ି ହ/ଶ 𝑎ସ/ହ𝑏ି ଻/ଷ (e) ඥ(−4𝑥)ଶ

ల

18. Perform the indicated operations and simplify if possible.

(a)

4 3 6

3 a  4 8 a (b) √6𝑏 − √24𝑏ଷ (c) ൫3 − √𝑥൯൫3 + √𝑥൯

(d) ൫2√𝑎 − 3√𝑏൯

ଶ

(e) 2 ඥ𝑦

య

൫4 ඥ𝑦

య

− 2 ඥ𝑦ଶ

య

൯ (f) ඥ18𝑦ଷ

య

√4𝑥ଶ

య

(g) 5ඥ16𝑦ସ

య

+ 7ඥ2𝑦

య

(h)

ඥ(௫ିଵ )య

ర

√௫ିଵ

19. A 16- foot tree casts a shadow that is 8 feet long. What is the distance from the

top of the tree to the end of the shadow? Leave answers in exact form.

20. David can paint the outside of a house in 12 hr. Bill can paint the same house in

9 hr. How long would it take them working together?

21. The current of the Gold River is 6 mph. A boat travels 50 mi downstream in the

same time that it takes to travel 30 mi upstream. What is the speed of the boat in

still water?

22. A toy rocket is shot vertically into the air from a launching pad 8 feet above the

ground with an initial velocity of 48 feet per second. The height h, in feet, of the

rocket above the ground at t seconds after launch is given by the function ℎ(𝑡) =

−16𝑡ଶ + 48𝑡 + 8.

How long will it take the rocket to reach its maximum height? What is the maximum

height?

23. A rectangular parking lot with a straight road as one side is to be fenced on the

other three sides by 1000 ft of aluminum fencing. If the area of the lot is to be

maximized, what should be its length and width? [Hint: write the area of the lot as

𝑎𝑥ଶ + 𝑏𝑥 + 𝑐.]

Solve the following equations. Include any complex solutions.

24. 11𝑥 = 2𝑥

####### ଶ

+ 12 25. 4𝑥

####### ଶ

= 8𝑥 26. 25𝑥

####### ଶ

− 9 = 0 27. 𝑥

####### ଶ

− 12 = 0

28. 𝑥ଶ − 10𝑥 − 3 = 0 29. 2𝑥ଶ + 1 = 4𝑥 30. 𝑦ଵ ଶ⁄ − 3𝑦ଵ ସ⁄ + 2 = 0

31. 𝑥଺ − 6𝑥ଷ + 5 = 0 32. 2𝑥ଶ − 6𝑥 + 3 = 0 (Hint: Use complete the square)

33. Use the intermediate value theorem to determine, if possible, whether the

function f has at least one real zero between a and b.

𝑓(𝑥) = 𝑥ସ − 6𝑥ଶ − 1; 𝑎 = 5, 𝑏 = 6

34. For the function 𝑓(𝑥) = 𝑥ଵହ − 2𝑥଼ + 7𝑥 − 3, state:

a) the maximum number of real zeros that the function can have;

b) the maximum number of x-intercepts that the graph of the function can have;

c) the maximum number of turning points that the graph of the function can have.

35. Determine visually whether the graph is symmetric with respect to the x-axis,

the y-axis, or the origin.

a. b.

46. For the triangle below, find the length of the side not given. Leave answers in

exact form.

(a) a = 8, b = 10 (b) a = √11, c = 5 (c) c = 1, b = x

Math 1021 Final Review– Answer Key

1. Not a function, one member of the domain corresponds to more than one

member of the range.

2. (a) 𝑓(4) = 2

(b) Domain: [-5,5]

(c) -2,

(d) Range: [-2,5]

3. (a) (−∞, ∞) (b) (−∞, 5) ∪ (5, ∞) (c) (−∞, ∞)

(d) (−∞, −7) ∪ (−7, −5) ∪ (−5, ∞) (e) (−∞,

####### ହ

####### ଷ

] (f) (0, ∞)

4. (a) 𝑥ଶ − 5𝑥 + 2 (b) 𝑥ଶ + 5𝑥 − 2 (c) 3,807 (d)

####### ି

####### ହ௫ାଶ

####### ௫మ

(e) −5𝑥

####### ଶ

+ 2 (f) 25𝑥

####### ଶ

− 20𝑥 + 4

5. (a) −12𝑥଻𝑦ଵଵ (b) 4𝑥ହ − 4𝑥ସ − 3𝑥ଷ + 14𝑥ଶ + 4𝑥 − 5

(c) 4𝑥ଶ + 12𝑥𝑦 + 9𝑦ଶ (d) 𝑥ସ − 16

(e) 4𝑥ଶ − 9𝑦ଶ (f) −8𝑥 ଼ + 12𝑥଻ − 4𝑥଺ − 20𝑥ହ

(g) 8𝑥ଷ − 36𝑥ଶ𝑦 + 54𝑥𝑦ଶ − 27𝑦ଷ

6. (a) (𝑥 − 5)ଶ (b) (2𝑥 + 3)ଶ (c)  3 x  2  3 x 2  (d)  2 x 1  3 x 5 

(e) (𝑥 − 2)(𝑥 + 6) (f) 2 x  x 2  x 2  2 x 4  (g)  2 x 3  x 1 

(h)  3 x  2  4 x 5  (i) 2𝑥𝑦(𝑦 − 8)(𝑦 − 3) (j) (9𝑥 − 2)ଶ

(k) 2𝑦(3𝑥ଶ − 2)(9𝑥ସ + 6𝑥ଶ + 4)

7. (a) 1 (b)

####### ଷ௫మି ହ௫ିସ

####### ଵଶ௫య

(c)

####### ଶ௬మି ଽ௬ି଺

####### (௬ିଶ )మ(௬ାଶ)

(d)

####### ଵ

####### ௬ାଵ

(e) −

####### ଵ

####### ௫

(f)

####### ଶ௫ିଵ

####### ௫ିଶ

(g)

####### ଵ

####### ଶ௫ାଷ௬

27. 𝑥 = ±2√3 28. 𝑥 = 5 ± 2√7 29. 𝑥 =

####### ଶ±√ଶ

####### ଶ

30. 𝑦 = 1,16 31. 𝑥 = √

య

, 1 32. 𝑥 =

####### ଷ±√ଷ

####### ଶ

33. Since f(a)=474 and f(b)=1079, f(a) and f(b) have same signs. Therefore, using

the intermediate value theorem, it cannot be determined whether the function f(x)

has a real zero between a=5 and b=6.

34. a. Has a maximum of 15 real zeros; b. has a maximum of 15 x-intercepts; c.

has a maximum of 15−1=14 turning points.

35. a. x-axis ; b. Origin.

36.

37. (−∞, 3) ∪ (7, ∞) 38. [−3, 1] 39.   4 , 2 

40. (−∞, −3] ∪ (−2,2) ∪ [4, ∞) 41. [−3, −1] ∪ [1, ∞)

42. (d); 𝑥 = 2, 𝑥 = −2, 𝑦 = 0 43. (a); 𝑥 = 2, 𝑥 = −2, 𝑦 = 8

44. (c); 𝑥 = 2, 𝑥 = −2, 45. (b);

46. a. 𝑐 = 2√41 b. 𝑏 = √14 c. 𝑎 = √1 − 𝑥ଶ

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