Summer Work for Students
Entering Honors PreCalculus
For the problems sets which start on page 7 write out all solutions clearly on a separate
sheet of paper. Show all steps and circle your answer.
The following are examples of different types of problems. Use these for reference
when working on the problem sets.
1) Exponents
Examples a)
(
3
= 3 ! 34 = 35 = 243
-4
3
b) 2 3 ! 32
( )
2) Example with variables 5x 4 x 2
) = ( 8 ! 9 ) = ( 72 )
2
2
2
= 5184
( )
3
= 5x 4 x 6 = 5x10
3) Radicals and rational exponents.
Example
a)
3
8 =2
'
! 27 $
c) # &
" 8%
d)
e)
3
3
2
24
4
3
b) 36 =
(
4
3
3
4
4
! 8 $
16
! 8$
! 2$
=# & =# 3
=# & =
&
" 27 %
" 3%
81
" 27 %
=
3
32a 4 b7 =
8(3 = 2 3 3
16 ( 2a 4 b7 = 4 a 2 b 3 2b1
4) Rationalizing Radicals
Examples
8 34
834 834
a) 3 ! 3 = 3
=
2
2
4
8
b)
)
36 = 63 = 216
= 43 4
2x
5+ 3
10x + 2x 3 5x + x 3
!
=
=
22
11
5- 3 5+ 3
5) Adding and Subtracting Radicals
Examples
a) 3 2 + 7 2 = 10 2
b) 2 50 - 12 8 = 10 2 - 24
6) Adding and Subtracting Polynomials
Examples
a) 2x 2 + 1 - x 2 - 2x + 1 = 2x 2 + 1 - x 2 + 2x - 1 = x 2 + 2x
(
(
) (
)
)
b) - 5z 3z3 - 4z = - 15z 4 + 20z 2
( x - 2) ( x 2 + 2x + 4 ) = x 3 + 2x 2 + 4x - 2x 2 - 4x - 8 = x 3 - 8
d) ( 3x - 5 ) ( 2x + 1) = 6x 2 - 10x + 3x - 5 = 6x 2 - 7x - 5
c)
1
2 = - 14 2
Summer Work for Students
Entering Honors PreCalculus
7) Factoring Polynomials
Examples
a) x 2 - 36 = ( x - 6 ) ( x + 6 )
b) x 2 - x - 6 = ( x - 3 ) ( x + 2 )
(
c) x 3 - x 2 + 2x - 2 = x 2 ( x - 1) + 2 ( x - 1) = ( x - 1) x 2 + 2
(
d) x 3 - 8 = ( x - 2 ) x 2 + 2x + 4
(
)
)
)
e) 3x - 12x - 36x = 3x x - 4x - 12 = 3x 2 ( x - 6 ) ( x + 2 )
4
3
2
2
2
8) Reducing Rational Expressions.
Examples
3xy
3xy
3y
a)
=
=
xy + x x ( y + 1) ( y + 1)
b)
9x 2 + 9x 9x ( x + 1)
9x
=
=
2x + 2
2 ( x + 1)
2
x 2 + 8x - 20
( x + 10) ( x - 2) = ( x - 2)
c) 2
=
x + 11x - 10
( x + 10) ( x - 1) ( x - 1)
9) Adding, Subtracting, Multiplying, and Dividing Rational Expressions
Examples
2 ( x + 3)
5(x - 4)
2
5
( 2x + 6) - 5 ( x - 4 ) =
a)
=
=
x - 4 x + 3 ( x + 3)( x - 4 ) ( x + 3)( x - 4 )
( x + 3)( x - 4 )
2x + 6 - 5x + 20
- 3 x + 26
=
( x + 3)( x - 4 )
( x + 3)( x - 4 )
x2 - x - 6
x 2 - 4 ( x - 3)( x + 2)
x+3
( x - 3) ! 1 =
b) 2
÷
=
!
=
x + 6x + 9 x + 3 ( x + 3 ) ( x + 3 ) ( x - 2 ) ( x + 2 ) ( x + 3 ) ( x - 2 )
( x - 3)
( x + 3)( x - 2)
10) Solving equations Linear, Rational, and Proportion
Examples
1
1
10
a) 2 ( x + 5 ) - 7 = 3 ( x - 2 )
b)
+
= 2
x-3 x+3 x -9
1
10 (
% 1
2x + 10 - 7 = 3x - 6
+
= 2 *
!"( x + 3 ) ( x - 3 ) #$ '
& x - 3 x + 3 x - 9)
2x + 3 = 3x - 6
9=x
( x + 3) + ( x - 3) = 10
2x = 10
x=5
c)
5x - 4 2
= ; 3 ( 5x - 4 ) = 2 ( 5x + 4 ) ; 15x - 12 = 10x + 8 ; 5x = 20 ; x = 4
5x + 4 3
2
Summer Work for Students
Entering Honors PreCalculus
11) Solving Quadratic Equations.
Examples
a) factoring x 2 - 2x - 8 = 0 ; ( x - 4 ) ( x + 2 ) = 0 ; x = 4, - 2
b) quadratic formula 2x 2 - 8x + 5 = 0 ; x =
8±
64 - 40
8 ± 24
8±2 6
; x=
;x=
;
4
4
4
4± 6
2
c) completing the square x 2 + 8x + 14 = 0 ; x 2 + 8x = - 14 ; x 2 + 8x + 16 = - 14 + 16 ;
x=
( x + 4 )2
( x + 4 )2
=2;
= ± 2 ; (x + 4) = ± 2 ; x = - 4 ± 2
12) Solving radical and absolute value equations
Examples
a) 3x + 10 - x = 0
b) 3 2x - 1 = 3
(
3x + 10 = x
(
3x + 10
)
2
= (x)
2
3
2x - 1
)
3
= ( 3)
c) 3x + 2 = 7
3
3x + 2 = 7 3x + 2 = - 7
2x - 1 = 27
3x + 10 = x 2
2x = 28
0 = x 2 - 3x - 10
x = 14
3x = 5
x=
3x = - 9
5
,-3
3
0 = ( x - 5)( x + 2)
x=5
(-2 is extraneous - does not work in the original equation)
13) Solve the following inequalities
Examples
a) x 2 + 4x - 5 > 0
( x + 5 ) ( x - 1) > 0
x
2
Divides number
line into
three sections
+ 4x - 5
x 2 + 4x - 5
= + this
section
x 2 + 4x - 5
= - this section
- 5
= + this section
0
3
1
Summer Work for Students
Entering Honors PreCalculus
13)
b) 2x + 14 + 3 ! 19
2x + 14 ! 16
-16 ! 2x + 14 ! 16
-30 ! 2x ! 2
-15 ! x ! 1
0
1
-15
14) Conversions from radians to degrees and degrees to radians. There are 2π radians
in one rotation and 360˚ in one rotation.
180 o
!
1 radian =
and 1 o =
radians
!
180
!
Conversions Examples
a) Convert
from radians to degrees
12
180!
180
" ! % " 180 %
$# 12 '& $# ! '& = 12! = 12 = 15˚
b) Convert -200˚ from degrees to radians:
"
!200"
!20"
!10"
=
=
( !200˚) #%$ &(' =
180
180
18
9
15) Coterminal Angles: angles of different measures with the same terminal side.
Examples
a) Find two different coterminal angles for 215˚
ß = 215˚ + 360˚(n)
Let n = 1, ß = 575˚
Let n = -1, ß = -145˚
b) Find two different coterminal angles for 8π/3
8"
8"
12 "
20 "
! =
+ 2 " ( n ) Let n = 2, ! =
+
=
3
3
3
3
8"
#12 "
# 4"
Let n =-2, ! =
+
=
3
3
3
4
Summer Work for Students
Entering Honors PreCalculus
16) Reference Angles: A reference angle is found by drawing an angle in Standard
Position and calculating the shortest distance back to the x-axis; REFERENCE
ANGLES ARE ALWAYS ACUTE!
Examples:
a) Find the reference angle for 327˚ ! = 360˚ - 327˚= 33˚
b) Find the reference angle for
7!
5
! =
7"
7"
5"
2"
# " =
#
=
5
5
5
5
17) Calculating sin, cos, tan of Major Angles in all quadrants.
Recall: sin = +/- sin(reference angle) depending on what quadrant the angle
lies in.
S ine
A ll
T an
C os
5
Summer Work for Students
Entering Honors PreCalculus
Examples: a) tan 135 0
0
RA!=!45 !
tan135 !=! ! tan 45 != !!1!
0
b) sin
0
!8 "
3
sin
6
−8π
−π
− 3
= − sin
= 3
3
2
Honors PreCalculus
Problem Set
Write out all solutions clearly on a separate sheet of paper. Show all steps and
circle your answer.
3
" 3% " 5 %
1) Simplify: a) $ ! ' $ '
# 5 & # 3&
2
(
2) Simplify: a) 6y 2y
2
b)
)
4 • 3-2 • 5 0
!2 !2 5 ( 3-1
(
)
b) ( x + 2 ) ( x + 2 )
4 2
3
!3
" 5x 2 %
c) $ -2 ' !
#y &
!1
3) Simplify:
a)
3
-27
e)
3
72x 3
4) Rationalize: a)
(
b)
f)
3
!125
3
)
3
c) 32
16
27
2
5
!
" 9%
d) $ '
# 4&
3
5
!
1
2
g) 4 32x 2 y 5
b)
3
4
3
c)
-3x
2- 7
5) Simplify:
a) 10 32 - 6 18
6) Simplify:
a) 5z - !" 3z - (10z + 6 ) #$
7) Factor:
a) 16x 2 - 9
c) 7 80x 3 + 2x 125x
b) 5 x - 3 x
b) ( 4x + 5 )
b) x 2 + x - 20
d) 2x 3 - x 2 - 6x + 3
18xy 2
8) Simplify: a)
3xy 2 + 6xy
2
(
)(
c) 3a 3 - 4b 2 3a 3 + 4b 2
)
c) x 3 + 125
e) 5x 5 - 10x 4 - 40x 3
3x 3 - 6x 2
b)
x2 - 4
9) Perform the indicated operation:
6
7
x-2
x 2 + 5x - 14
a)
+
b)
÷
x +1 x - 8
x2 - 4
x+2
7
x 2 - 7x + 12
c) 2
x + 3x - 18
Honors PreCalculus
Problem Set
10) Solve the following equations:
a) 2 (13x - 15 ) + 3 ( x - 19 ) = 0
c)
b)
1
3
4
+
= 2
x- 2 x+3 x +x-6
10x + 3 1
=
5x + 6 2
11) Solve the following quadratic equations by:
a) factoring x 2 + 11x - 26 = 0
b) quadratic formula 4x 2 - 4x - 4 = 0
12) Solve each of the following:
a) 2x - 15 - 4x = 0
b)
3
3x + 1 = 5
13) Solve for the inequalities:
a) 3 - 2 x < 13
b) x 2 - 6x - 7 < 0
c) 2x - 4 = 10
c)
5x + 21 + 4 ! 30
14) Graph each of the following – do not use your calculator. State the domain and
range of each.
#2x!+!3,!x !!"2
a) ! f ( x )!=! $ 2
!!!!!!!!!!!!!!!!!!!!!!!!!!b)!y!=! x!"!3!+!2!!!!!!!!!!!!!
% x !"!1,!x!>!"2
c)!y!=!x ( x!+!1) ( x!"!2 ) ( x!"!5 )!!!!!!!!!!!!!!!!!!d)!y!=!
2
x!"!4
!!!!!!!!!!!!!!!e)! f ( x )!=!"! x!+!2 !
x!+!2
15) Find the inverse of each of the following.
a) !y!=!3x!!!14!!!!!!!!!!!!!!!!b)!y!=! x!!!3!+!2!!!!!!!!!!!!c)!!y!=!
x!!!4
!!!!!!!!!!d)!y!=!x 3 !!!5!
x!+!2
16) Evaluate each series, if the sum exists.
k
8
*
*
$ 6'
$ 5'
a) ! " ( 4k !!!7 )!!!!!!!!!!!!!!!!!!!!!!!!!!b)! " 13# & ) !!!!!!!!!!!!!!!!!!!!!!!c)! " !4 & )
% 5(
% 7(
k ! = !1
k ! = !1
n ! =1
n ! ! !1
17) Find the first term and common difference of the arithmetic sequence and write a
recursive definition:
4th term is 3; 20th term is 35
18) Find x so that x,!x!+!2,!and x!+!3! are consecutive terms of a geometric sequence.
19) Solve each of the following – no calculators
1
a)!log x 216!=!3!!!!!!!!!!b)!e x + 1 !=! !4 !!!!!!!!!!!!!!c)!log 3 x!+!log 3 ( x!!!6 )!=!3!
e
8
Honors PreCalculus
Problem Set
20) Sue invested $2560 in an account that pays 4.38%. Find the value of her account
after 5 years if a) the money is compounded monthly, and b) it is compounded
continuously.
21) Convert the following polar coordinates to rectangular coordinates.
a)!(2, 210 0 )!!!!!!!!!!!!!!!!!!!!!!!!!b)! 2.4,!78 0 !
(
)
22) Convert the following rectangular coordinates to polar coordinates.
a)!(8,!!15)!!!!!!!!!!!!!!!!!!!!!!!!!b)!( !3.5,!!10.5 )!!
(
)
(
)
23) Let t !=!( 3,!2 ) ,!u!=!( !4,!5 ) ,!v!=! 3,!270 0 ,!and w!=! 5,!120 0 ! . Write the resultant vector
in rectangular and polar form. Give decimal answers to the nearest hundredth.
a)!t + u
b)! u!!!2t !!!!!!!!!!!!!c)!2v!!!w!!!!!!!!!!!!!d)!v!+!w!
24) Complete the following graph by labeling the major angles in both radians and
degrees:
90 o =
60 o =
45 o =
30 o =
!
6
0o = 0
25) Find two different coterminal angles for each of the following.
3!
a) 112 o
b)
c) " 21o
5
26) Find the reference angle for each of the following.
5"
!11"
a) 25 o b) ! 301o
c)
d)
8
4
9
Honors PreCalculus
Problem Set
27) Complete the following table using the unit circle – no calculators.
Degrees
00 !
30 0 !
45 0 !
60 0 !
90 0 !
180 0 !
270 0 !
Radians
Sin
Cos
tan
360 0 !
28) Solve the following right triangle:
b
z
15
x
20 o
29) A 12-meter flagpole casts a 9-meter shadow. Find the angle of elevation of the
sun.
10