Math 101: College Algebra Print Name:

Fall 2018

Final Exam Signature:

Time Limit: 3 Hours

No electronic devices or calculators allowed. Show all of your work on this test paper. Putanswers in the blanks provided. This exam has 33 questions worth 200 points. The pointvalues of each question are shown in the left margin in [brackets].

1.[4] Factor completely. If the polynomial cannot be factored, say it is prime.

x3 � 8

1.(x� 2)(x2 + 2x+ 4)

2.[4] Factor completely. If the polynomial cannot be factored, say it is prime.

64y2 � 25

2.(8y � 5)(8y + 5)

3.[4] Factor completely. If the polynomial cannot be factored, say it is prime.

3x2 + 5x� 2

3.(x+ 2)(3x� 1)

4.[4] Simplify the expression.9 3p24� 3

p81

4.15 3

p3

5.[4] Simplify the radical. p720

5.12p5

20 pts A pg 1 of 10

Solutions

For mu la :

a3

- b3

= ( a- b ) ( a

't a b t b2)

= ( 8 y )2

- (5)2

Formula :

a2

- b 2= ( a t b ) C a

- b )

3×2 - 6 x t I x - 2

3 x ( x - 2 ) t I ( x - 2)

( 3 x t1) ( x - 2)

9 ] 2.2.27 - ]3.3.33- =

9. 295 - 385'

I 805 - 305

-

✓ 72T = ✓ 2 . 2 . 2 . 2 . 3 . 3 . 5

720 = 4 - 3 TS'

10^728^9

6.[10] Simplify the expression. Factor any polynomial appearing in your answer.

1

2+

3

xx+ 3

4

6.

2(x+ 6)

x(x+ 3)

7.[6] Simplify the expression. Express your answer so that only positive exponents occur.✓

9x2y

x1/3y5

◆�1/2

7.

y2

3x5/6

8.[6] Solve the equation.x

x+ 2=

3

2

8.x = �6

22 pts A pg 2 of 10

¥2X

=-

E+z=¥¥= 't ¥2

= . ¥3=

2 ( x -16 )

¥33

=t¥)"a

= Cats ,"

2

y= -

3×912• 12

2X =3 ( x -12 )

2x = 3×+6

- Ix = 6

X = -6

9.[6] Solve the equation.(5w � 2)1/3 = 2

9.w = 2

10.[6] Solve the equation.6x� 5 =

6

x

10.x = �2

3 , x = 32

11.[6] Solve the equation. ���2y + 3��� = 5

11.y = �4, y = 1

18 pts A pg 3 of 10

¢Sw - 2)"

3) 3=23

5W -2=8

5W = IO

w -_ ¥

(6×-5)×-6

6×2-5×-6 2x -3=0,

3×+2=0

6×2-5×-6=0 2x=3 3X= -2

( 2x - 3) (3×+2)=0 ×=z X= I

Ly -13=5, 2y -13=-5

2y=2 2y= -8

y=I y=-4

12.[6] Solve each inequality and write your answer using interval notation.���1� 4x

��� < 5

12.

��1, 32

�

13.[6] Find the distance between the points A(4,�3) and B(6,�7) and simplify your answer.

13.2p5

14.[8] Consider the equation x2 � y � 4 = 0.(a) Find all x-intercepts on the graph. If none, write “none.”

(a)x = 2,�2 or (±2, 0)

(b) Find all y-intercepts on the graph. If none, write “none.”

(b)y = �4 or (0,�4)

15.[5] Multiply the complex numbers and write your answer in standard form a± bi:

(8 + 4i) · (5 + 5i)

15.20 + 60i

25 pts A pg 4 of 10

- 5h I - 4 x L 5

- 6 s - 4×24

¥4 > x > - I

I > x > I

-

✓ ( -7 - C-3) Yt ( 6-45 =

Vt4TtCzT=Vl6tT =

1207×2-0-4=0X

'

-4=0

( x- 2) ( x -123=0

O'

- y- 4=0

- y-

- 4

y = -4

= 40+40 it

205+205=40+60: -20

16.[6] Find all real and complex solutions:

x2 + 12x+ 40 = 0

16.x = �6± 2i

17.[5] Find the polynomial of smallest degree whose graph matches the graph displayed below.Circle your answer.

x

y

(�1, 0) (2, 0)(0, 0)

A. y = 2x(x+ 2)2(x� 1)

B. y = �2x(x+ 2)2(x� 1)2

C. y = �32x(x� 2)2(x+ 1)

D. y = �3x2(x� 2)(x+ 1)2

E. y = 23x

3(x� 2)2

F. None of these

18.[6] Find the slope-intercept equation of the line passing through (�3, 6) and (2, 1).

18.y = �1x+ 3

17 pts A pg 5 of 10

x =

- 12 ±Vl44-4t2-

× = -12+-1-162 = -12124hL = -12g ±4÷

Factors :

( x -11 ),

x,Cx - 25 X

X

X

x

m = 9×37×4 -2¥ = = - I

y = mxtb

6 =L - 1) f 3) + b

6 =3 t b

b =3

19.[6] Solve the system of linear equations:(

2x+ 3y = 1

�x+ y = �3

19.x = 2, y = �1

20.[8] Complete the square to find the radius and the center of the circle with equation

x2 + y2 � 6x+ 4y = 12

Center: (3,�2) Radius: 5

21.[6] Determine the domain of the function f(x) =p3� 6x and write your answer using

interval notation.

21.

��1, 12

⇤

20 pts A pg 6 of 10

{2X +3g =/

- 2x t2y= -6-

-

Sy = -5

y = -

I

2×+31-17=1

2×-3=12x=4

X=2

X' -6×+9-+92-14g-1-4=12+9-+42

( x - 3)'

t ( y -1272=25

Rule : radicand ZO

3-6×20

-6×2-3

XE =L

22.[8] The monthly cost C, in dollars, for cell phone calls from the United States to Canadaon a popular phone plan is modeled by the function C(x) = 0.15x + 4, where x is thenumber of minutes used.(a) What is the cost of using 100 minutes?

(a)$19

(b) If the bill was $7, how many minutes were used?

(b)20 minutes

23.[6] Find the equation of the quadratic function with vertex (1,�5) and y-intercept is �3.

23.y = 2(x� 1)2 � 5

14 pts A pg 7 of 10

C ( 1007=0.15400 ) +4=151-4--19

7=0.15×-14

3=0.15 X

⇐ = 3%1=20 = X

-

y -_ a ( x - h )'

+ k ( O ,-3 )

y=a( x - D'

-5

-3 = a (0-1)--5-3 -

- aft )'

-5

-3 -

- a -5

2 = a

24.[8] The graph below displays a linear function y = g(x) and a quadratic function y = f(x).Use the graph to answer the questions.

y = f(x)y = g(x)

(4,�3)

(6, 1)(2, 1)

(2, 3)

(4, 2)

x

y

�4

�3

�2

�1

1

2

3

4

5

�1 1 2 3 4 5 6

(a) When is g(x) = 1?

(a)

(b) Find (g � f)(4)

(b)

(c) Find⇣

fg

⌘(4)

(c)

(d) Find (f � g) (4)

(d)

25.[6] Graph the line 3x� 4y + 12 = 0 and label any intercept that appears on your graph.

x

y

�5

�4

�3

�2

�1

1

2

3

4

5

�5 �4 �3 �2 �1 1 2 3 4 5

26.[8] Suppose R(x) =3x+ 5

x� 6.

(a) Write the equation of any horizontal asymptote appearing on the graph of y = R(x).If none exist, write “none.”

(a)y = 3

(b) Write the equation of any vertical asymptote appearing on the graph of y = R(x).If none exist, write “none.”

(b)x = 6

22 pts A pg 8 of 10

It 6

gc 4) =3

f (4) =-324-3 ) g

-32

f- ( gas )=f( 2) I

7 3×-49=-12x - int : ( -4,0 )

•( 0,3 ) y

- int : ( 0,3 )C- 4,0 )

yBe

27.[8] Solve the inequality below using a Sign Chart. Display your Sign Chart on the numberline. Give the solution set using interval notation.

x� 3

x+ 1� 0

�10�9 �8 �7 �6 �5 �4 �3 �2 �1 0 1 2 3 4 5 6 7 8 9 10

27.(1,�1) [ [3,1)

28.[6] Suppose that f(x) = x2 + 3x� 1 and g(x) = 2x+ 3. Find the composite function f � gand write the answer in standard form.

28.4x2 + 18x+ 17

29.[8] For the function f(x), find the inverse function f�1(x):

f(x) =4

x� 2

29.f�1(x) =

4

x+ 2

22 pts A pg 9 of 10

Zeros : X =3, X= - I

l!

I=¥18¥ SITE

ii

⑦ i ① i ④• . •

'•

I I

t I

l I

I

( fog) ( x ) = f (2×+3)=(2×-135+312×+3) -

1=4×76×+6×+9-16×+9-1=4×2+18×+17

y=¥ -24

X -

- f - 2 y= ¥

X -12=5

yCx -127=4

30.[4] Simplify:✓

1

100

◆�3/2

30.1000

31.[4] Convert 2rk = P to logarithmic form.

31.log2(P ) = rk

32.[8] Solve the equation:252x = 5x

2�12

32.�2, 6

33.[4] Find the exact value of log2 (64).

33.6

20 pts A pg 10 of 10

= ( 1003312=410073

= 103=1,000

dog !2"

)=logfP )

rk -

- dogs ( P )

( 574=5×2-12

5415×2-120=6-6 )(xt2 )

4x=x'

-12X -6=0

,X -12=0

0=15-4×-12

log .cat/=IogzC26)--6