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Honors Pre‐Calculus: 2.3: Polynomial Functions of Higher Degree with Modeling
A. Sketch a graph of each of the following functions
1. ( )( )( )( ) 3 2 1 4f x x x x= + − − 2. 4 2( ) 4f x x x= −
3. ( )( )2( ) 2 1f x x x x= + − 4. ( ) ( )2( ) 2 3f x x x= − +
B. Match the following functions with the corresponding graphs among a‐f
5. 3( )f x x= a. b.
c.
6. 4( ) 3f x x= +
7. ( ) ( 1)( 2)f x x x x= + −
8. 3( ) 3 1f x x x= − + e.
f.
g.
9. 4( )f x x= − 10. ( )( )2 2( ) 4 9f x x x= − −
C. List the end behavior of the following functions using limits
11. 4 3( ) 2 4 2f x x x x= − + + − 12. ( ) ( ) ( )2 3( ) 2 1 2 4 5f x x x x= − − − −
40
30
20
10
-10
-20
-30
-4 -2 2 4
6
4
2
-2
-4
-6
-5 5
4
2
-2
-4
5
15
10
5
-5
5
2
-2
4
2
-2
2
-2
2
-2
30
20
10
2
Honors Pre‐Calculus 2.4 Worksheet
A. Find a cubic polynomial in standard form through the point (‐1,‐20) with zeros ‐2, ‐1/3, and 3/2.
B. Consider the function 4 3 2( ) 2 5 13 25 15f x x x x x= + − − +
1. List all possible rational roots.
2. Show that all of the roots are in the range (‐4,3).
3. Use your calculator to find the rational roots, and then verify them using the remainder theorem.
4. Use synthetic division to find all roots. Then, write the polynomial in factored form.
Honors Pre‐Calculus 2.5‐2.6 Worksheet Halldorson
A. Answer each of the following for 1 3 2z i= − and 2 2z i= − +
1. 1 23 2z z−
2. 1 2z z⋅
3. 1
2
zz
4. 1 2z z+
5. ( )21z
6. ( )42z
7. 1z
8. 2z
9. The midpoint of 1 2 and z z
10. The distance between 1 2 and z z
11. Plot the points on the complex plane below:
4
2
-2
-4i
i
i
i
Honors Pre‐Calculus 2.5‐2.6 Worksheet Halldorson
B. Evaluate each of the following:
1. 15i
2. 592i 3. 7i− 4. 34i−
C. Determine the nature of the roots, then solve each quadratic equation
1. 2( ) 8 15f x x x= + − 2. 2( ) 3 4 9f x x x= − +
D. Write a polynomial of least degree in standard form with the following zeros
1. ‐2 and 1‐2i 2. and 1i i− +
E. The polynomial 3 2( ) 3 8 30f x x x x= − − + has 3+i as a root. Find all roots, and write its linear factorization.
Honors Pre-Calculus 2.7 Worksheet A.
Write how each function can be obtained from the graph of 1
( )f xx
with transformations.
1. 4
( )1
g xx
2. 2 1
( )4
xg x
x
B. For each of the following functions, find the end behavior asymptote and all vertical asymptotes. Also write the end behavior in limit notation.
3. 2
2
6 8( )
4 3
x xf x
x x
4. 3 22 4 18( )
2
x x xf x
x
5. 3 2
2
2 3( )
2
x x xf x
x x
6. 2
2( )
6
xf x
x x
C. Sketch a graph of each of the following functions. Sketch all asymptotes and label all intercepts. 7. 3
2
2 1( )
1
xf x
x
8. 2
2
3 8 5( )
2 9 7
x xf x
x x
8
6
4
2
-2
-4
-6
-8
-10
-10 -5 5 10
10
5
-5
-10
-10 10
Solving Rational Equations ©2001-2003www.beaconlearningcenter.com Rev.7/25/03
SOLVING RATIONAL EQUATIONS EXAMPLES
1. Recall that you can solve equations containing fractions by using the least common denominator of all the fractions in the equation. Multiplying each side of the equation by the common denominator eliminates the fractions. This method can also be used with rational equations. Rational equations are equations containing rational expressions.
2. Example: solve4
4−x + 3x = 6.
44−x +
3x = 6.
)6(12)34
4(12 =+− xx
3(x – 4) + 4(x) = 72 3x – 12 + 4x = 72 7x = 84 x = 12
The LCD of the fraction is 12.
Multiply each side of the equation by 12. The fractions are eliminated.
Emphasize that each term must be multiplied by the LCD in order to have a balanced equation. A common mistake is to multiply only those terms that are expressed in fractions.
Check 634
4=+
− xx 63
124
412=+
− 2 + 4 = 6 6 = 6
Solving Rational Equations ©2001-2003www.beaconlearningcenter.com Rev.7/25/03
3. Example: solve 21
223
−=+
−x
xx
4. Example: Solve 353
1=−
+ kk .
21
223
−=+
−x
xx
)2)(1(2)1
223)(1(2 −+=
+−+ xx
xx
xxx
xxxxx 44)2(2)1(3 2 −−=−+ xxxx 44433 22 −−=−+
7x = -3
73
−=x
Note that x ≠ -1 and x ≠ 0. The LCD of the fractions is 2x(x + 1)
Multiply each side of the equation by 2x(x + 1).
Check 2)1
73(
)73(2
)73(2
3−=
+−
−−
−
2
7476
76
3−=
−−
−
223
621
−=+− 269
621
−=+−
26
12−=− -2 = -2
353
1=−
+ kk
)3(15)5
(15)3
1(15 =−+ kk
5(k + 1) – 3(k) = 45 5k + 5 – 3k = 45 2k + 5 = 45 2k = 40 k = 20
Multiply by each side by the LCD which is “15”.
Check 3520
3120
=−+
3520
321
=−
7 – 4 = 3 3 = 3
Solving Rational Equations ©2001-2003www.beaconlearningcenter.com Rev.7/25/03
5. Example: Solve41
196
=−
−xx
“x” cannot equal “0” or “1”.
Multiply each side of the equation by the LCD whish is 4x(x – 1) 4
11
96=
−−
xx
41)1(4
19)1(46)1(4 −=−
−−− xxx
xxx
xx
4(x – 1)6 – 4x(9) = x2 – x 24x – 24 – 36x = x2 – x 0 = x2 + 11x + 24 0 = (x + 3)(x + 8) x = -3 or - 8
Check41
139
36
=−−
−−
41
492 =−
−−
41
4122 =+−
41
41=
Check41
189
86
=−−
−−
41
99
43
=+−
411
43
=+−
41
41=
Solving Rational Equations ©2001-2003www.beaconlearningcenter.com Rev.7/25/03
6. Example: solvex
xx
x−−
=−
−3
13
2
xx
xx
−−
=−
−3
13
2
)3(1
32
−−−
=−
−x
xx
x
)3(1
32
−−
−=−
−xx
xx
)3()3(
13
2)3()3( −−−
−=−
−−− xxx
xxxx
x(x – 3) – 2 = – (x – 1) x2 – 3x – 2 = -x + 1 x2 –2x – 3 = 0 (x – 3)(x + 1) = 0 x = 3 or x = -1 Since “x” cannot equal 3, the only solution is x = -1
“x” cannot equal 3
Multiply both sides of the equation by the LCD which is “x – 3”
Have students name the restrictions on the domain of an equation before solving it. Emphasize the importance of this when determining the solutions for an equation. In this example, the domain does not include 3. This limits the solutions to only –1.
Check1311
312)1(
−−−−
=−−
−−
21
211 −=+−
21
21
−=−
Solving Rational Equations ©2001-2003www.beaconlearningcenter.com Rev.7/25/03
7. Example: solve 115
12
2 =−−
+− m
mm
m
Check: 1)1)(1(
51
2=
−+−
+− mm
mm
m 1)14)(14(
5414)4(2
=−−+−
−−+
−−−
)5)(3(9
58
−−−
+−− = 1
159
1524
− = 1 11515
=
1)1)(1(
51
2=
−+−
+− mm
mm
m
)1)(1)(1()1)(1(
5)1)(1(1
2)1)(1( −+=−+
−+−+
−+− mm
mmmmm
mmmm
2m(m + 1) + (m – 5) = m2 – 1 2m2 + 2m + m – 5 = m2 – 1 m2 + 3m – 4 = 0 (m + 4)(m – 1) = 0 m = -4 or 1 Since “1” cannot be a solution then “m” must equal “-4”
m cannot equal 1 or –1.
Solving Rational Equations ©2001-2003www.beaconlearningcenter.com Rev.7/25/03
SOLVING RATIONAL EQUATIONS WORKSHEET Solve each equation and check (state excluded values).
1. 21
32
632
+=− aa
2. 14
327
32 +=−
− bbb
3. 127
53
=+xx
4. 522
5=+
+ kkk
5. 11
51
=−
++ mm
m
6. 223
223
4=
++
− xx
xx
7. 255
5 2
−=−
−− p
pp
8. 3
122332
+=−
−−
aaa
9. 2
32252
+=−
−−
bbb
10. 6
12128
42 −
+−
=+− kk
kkk
Name:____________________ Date:____________ Class:____________________
Solving Rational Equations ©2001-2003www.beaconlearningcenter.com Rev.7/25/03
SOLVING RATIONAL EQUATIONS WORKSHEET KEY Solve each equation:
1. 21
32
632
+=− aa
2. 14
327
32 +=−
− bbb
3. 127
53
=+xx
)21(6)
32(6)
632(6 +=
− aa
2a – 3 = 2(2a) + 3(1) 2a – 3 = 4a + 3 -3 = 2a + 3 –6 = 2a –3 = a
Check:
21
3)3(2
63)3(2
+−
=−−
212
23
+−=−
211
211 −=−
)14
3(14)2
(14)7
32(14 +=−
− bbb
2(2b – 3) – 7(b) = 1(b + 3) 4b – 6 – 7b = b + 3 -9 = 4b
b=−49
Check:
14
349
249
7
3)49(2 +
−
=
−
−−
−
14
349
249
7
329
+−
=
−
−−
−
1443
249
7215
=
−
−
−
563
89
1415
=+−
563
563=
)1(10)27(10)
53(10 x
xx
xx =+
2(3) + 5(7) = 10x 6 + 35 = 10x 41 = 10x
x=1041
“x” cannot equal “0”
Check 1)
1041(2
7
)1041(5
3=+
1
10827
102053
=+
18270
20530
=+
1 = 1
Solving Rational Equations ©2001-2003www.beaconlearningcenter.com Rev.7/25/03
4. 522
5=+
+ kkk
5. 11
51
=−
++ mm
m
5)2(2)2()2(1
5)2( +=+++
+ kkk
kkk
kkk
5k2 + 2k + 4 = 5k2 + 10k 2k + 4 = 10k 4 = 8k
k=21
“k” cannot equal “-2” or “0”
5
212
221
)21(5
=++
54
2525
=+
5 = 5
1)1)(1()1(1
5)1)(1()1(1
)1)(1( −+=−
−+++
−+ mmm
mmmmmm
(m – 1)m + (m + 1)(5) = (m + 1)(m – 1) m2 – m + 5m + 5 = m2 – 1 4m + 5 = –1 4m = -6
23
−=m
Check 11
235
123
23
=−−
++−
−
1
255
2123
=−−
−
6 – 5 = 1 1 = 1
“m” cannot equal “-1” or “1”
Solving Rational Equations ©2001-2003www.beaconlearningcenter.com Rev.7/25/03
6. 223
223
4=
++
− xx
xx
7. 255
5 2
−=−
−− p
pp
2)23)(23()23(1
2)23)(23()23(1
4)23)(23( +−=+
+−+−
+− xxx
xxxx
xxx
(3x + 2)4x + (3x – 2)2x = 18x2 – 8 12x2 + 8x + 6x2 – 4x = 18x2 – 8 4x = –8 x = –2
Check 22)2(3
)2(22)2(3
)2(4=
+−−
+−−
−
244
88
=−−
+−−
1 + 1 = 2 2 = 2
“x” cannot equal 32 or
32
−
255 2
−=−−
pp
)2)(5()5(1
5)5(2
−−=−−
− pp
pp
5 – p2 = -10 + 2p 0 = p2 + 2p – 15 0 = (p + 5)(p – 3) p = -5 or 3
Check 255
)5(55
5 2
−=−−
−−
−−
225
21
−=− -2 = -2
Check 23535 2
−=−−
224
−=−
-2 = -2
“p” is not equal to “5”
Solving Rational Equations ©2001-2003www.beaconlearningcenter.com Rev.7/25/03
8. 3
122332
+=−
−−
aaa
9. 2
32252
+=−
−−
bbb
)3(112)3)(3(2)3)(3(
)3(132)3)(3(
++−=+−−
−−
+−a
aaaaaaaa
(a + 3)(2a – 3) – 2a2 + 18 = (a – 3)(12) 2a2 + 3a – 9 – 2a2 + 18 = 12a – 36 3a + 9 = 12a – 36 45 = 9a 5 = a
Check35
12235
3)5(2+
=−−−
8122
27
=−
23
23=
“a” cannot equal “3” or “-3”
)2(13)2)(2(2)2)(2(
)2(152)2)(2(
++−=+−−
−−
+−b
bbbbbbbb
(b + 2)(2b – 5) –2b2 + 8 = (b – 2)3 2b2 –b – 10 – 2b2 + 8 = 3b – 6 -b – 2 = 3b – 6 4 = 4b 1 = b
Check21
3221
5)1(2+
=−−−
332
13
=−−−
1 = 1
“b” cannot equal “2 or “-2”
Solving Rational Equations ©2001-2003www.beaconlearningcenter.com Rev.7/25/03
10. 6
12128
42 −
+−
=+− kk
kkk
)6(11)6)(2(
)2(1)6)(2(
)6)(2(4)6)(2(
−−−+
−−−=
−−−−
kkk
kkkk
kkkk
4 = (k – 6)k + (k – 2)1 4 = k2 – 6k + k – 2 0 = k2 – 5k – 6 0 = (k – 6)(k + 1) k = 6 or –1 Since “k” cannot equal “6” the solution is “-1”
Check61
121
112)1(8)1(
42 −−
+−−−
=+−−−
71
31
214
−=
214
214=
“k” cannot equal “2” or “6”
Solving Rational Equations ©2001-2003www.beaconlearningcenter.com Rev.7/25/03
SOLVING RATIONAL EQUATIONS CHECKLIST
1. On question 1, did the student solve the equation correctly and check solutions? a. Yes (20 points) b. Solved equation correctly but did not check solutions (15 points) c. Equation was solved incorrectly but had only minor mathematical errors.
Student did check solutions (10 points) d. Equation was solved incorrectly and student did not check solutions (5
points)
2. On question 2, did the student solve the equation correctly and check solutions? a. Yes (20 points) b. Solved equation correctly but did not check solutions (15 points) c. Equation was solved incorrectly but had only minor mathematical errors.
Student did check solutions (10 points) d. Equation was solved incorrectly and student did not check solutions (5
points)
3. On question 3, did the student solve the equation correctly and check solutions? a. Yes (20 points) b. Solved equation correctly but did not check solutions (15 points) c. Equation was solved incorrectly but had only minor mathematical errors.
Student did check solutions (10 points) d. Equation was solved incorrectly and student did not check solutions (5
points)
4. On question 4, did the student solve the equation correctly and check solutions? a. Yes (20 points) b. Solved equation correctly but did not check solutions (15 points) c. Equation was solved incorrectly but had only minor mathematical errors.
Student did check solutions (10 points) d. Equation was solved incorrectly and student did not check solutions (5
points)
5. On question 5, did the student solve the equation correctly and check solutions? a. Yes (20 points) b. Solved equation correctly but did not check solutions (15 points) c. Equation was solved incorrectly but had only minor mathematical errors.
Student did check solutions (10 points) d. Equation was solved incorrectly and student did not check solutions (5
points)
Student Name: __________________ Date: ______________
Solving Rational Equations ©2001-2003www.beaconlearningcenter.com Rev.7/25/03
6. On question 6, did the student solve the equation correctly and check solutions? a. Yes (20 points) b. Solved equation correctly but did not check solutions (15 points) c. Equation was solved incorrectly but had only minor mathematical errors.
Student did check solutions (10 points) d. Equation was solved incorrectly and student did not check solutions (5
points)
7. On question 7, did the student solve the equation correctly and check solutions? a. Yes (20 points) b. Solved equation correctly but did not check solutions (15 points) c. Equation was solved incorrectly but had only minor mathematical errors.
Student did check solutions (10 points) d. Equation was solved incorrectly and student did not check solutions (5
points)
8. On question 8, did the student solve the equation correctly and check solutions? a. Yes (20 points) b. Solved equation correctly but did not check solutions (15 points) c. Equation was solved incorrectly but had only minor mathematical errors.
Student did check solutions (10 points) d. Equation was solved incorrectly and student did not check solutions (5
points)
9. On question 9, did the student solve the equation correctly and check solutions? a. Yes (20 points) b. Solved equation correctly but did not check solutions (15 points) c. Equation was solved incorrectly but had only minor mathematical errors.
Student did check solutions (10 points) d. Equation was solved incorrectly and student did not check solutions (5
points)
10. On question 10, did the student solve the equation correctly and check solutions? a. Yes (20 points) b. Solved equation correctly but did not check solutions (15 points) c. Equation was solved incorrectly but had only minor mathematical errors.
Student did check solutions (10 points) d. Equation was solved incorrectly and student did not check solutions (5
points)
Solving Rational Equations ©2001-2003www.beaconlearningcenter.com Rev.7/25/03
Total Number of Points _________ A 180 points and above B 160 points and above C 140 points and above D 120 points and above F 119 points and below
Any score below C needs remediation!
Honors Pre‐Calculus 2.9: Rational Inequalities Halldorson
Solve each inequality. Show all work, and give your answer on a number line and in interval notation.
1. 3 22 5 14 8 0x x x+ − − > 2. 32 27 3 (6 )x x x+ ≤ +
3. x
x x>
+1
2 4. x
x<
4
5. x
x+
≤−
2 1 05
6. +
≥−
xx
3 13