Final Exam Review, Unit 1

Report
Make sure you know the
day and time of the final exam for
this section of Math 110:
Day: ______ Date:______
Time: ______ to _______
• All Math 110 finals will be given in
your regular classroom.
• Final exam schedules for all sections
are posted on the bulletin board and in
your course syllabus.
Please
CLOSE
YOUR LAPTOPS,
and turn off and put away your
cell phones,
and get out your notetaking materials.
Math 110 Final Exam:
•
•
•
•
•
Comprehensive – covers the whole semester
Worth 200 points (20% of course grade)
47 questions
Test period is one hour and 50 minutes
Practice Final is worth 10 points and also has 47 questions
and unlimited tries.
• Your best score on the practice final will also earn up to
20 extra credit points on the final.
• The practice final is due at the scheduled start time of this
section’s final exam.
• IMPORTANT NOTE!!! To get access to the practice final, you
must have submitted all four final exam review homework
assignments and have scored at least 80% on each one.
Final Exam Review Materials:
1. Four review homework assignments:
These are each
worth 15 points, vs. the usual homework assignment of 4 points.
• Unit 1 review homework (Chapters 1, 2, 3).
Due today. (If you need an extension see or email me TODAY
so you can finish it and get access to the Unit 2 review HW.)
• Unit 2 review homework (Chapters 4, 5, 8):
NOTE: You must get at least 80% on the Unit 1 Review before you can start
the Unit 2 review. Due at the start of the class on Monday.
• Unit 3 review homework (Chapter 6):
NOTE: You must get at least 80% on the Unit 2 Review before you can start
the Unit 3 review. Due at the start of the class on Tuesday..
•Unit 4 review homework (Chapter 7):
NOTE: You must get at least 80% on the Unit 3 Review before you can start
the Unit 4 review. Due at 8:00 p.m. on Study Day (Wed., May 8)
Final Exam Review Materials:
2. Comprehensive Practice Final
NOTE: You must get at least 80% on the Unit 4 Review Homework before
you can start the comprehensive practice final.
•
•
The practice final has 47 questions, as does the regular
final exam. The questions will NOT be exactly the same as
those on the regular final, but they will be similar in content
and are distributed across sections the same way as on the
regular final.
Each time you take the practice final you will get a different
set of questions, AND your highest score will earn you that
percentage of 20 extra credit points on the final, so it will
pay off if you can find the time to take it more than once.
Comprehensive Practice Final:
Once you finish the Unit 4 Final Review homework with at least
an 80% score, you will have access to the practice final, with as
many attempts as you want.
The practice final is worth 10 points, plus a possible bonus of
up to 20 extra credit points on the final exam. Your best
percentage score on the practice final will earn that percent of
10 points toward your course grade, plus that best score percent
times 20 extra credit points will be added on to your final exam
point score before partial credit points are added.
This makes it extremely important for you to make sure you work
to get at least 80% on each of the four final exam review
homework assignments so that you can get access to the
practice final.
Math TLC Open Lab Hours:
Room 203
Jarvis Hall Science Wing
Regular hours through Tuesday, May 7 (8 AM – 6:30 PM)
Finals Week Open Lab Hours:
Starting at 10 a.m. on Study Day (Wed.)
and 8 a.m. on Final Exam days
See schedule posted in the classroom
and in the open lab and online at
http://www.uwstout.edu/mathtlc
Final Exam Review,
Unit 2
Unit 2 Covers:
Section 4.1
(Systems of linear equations in 2 variables)
Chapter 5
(Properties of exponents, operations on polynomials,
factoring polynomials, solving polynomial equations)
Section 8.2
(Quadratic formula)
Section 4.1 Review:
Example
Solve the following system of equations using the
elimination method.
6x – 3y = -3 and 4x + 5y = -9
•
Multiply both sides of the first equation by 5 and the
second equation by 3.
First equation,
5(6x – 3y) = 5(-3)
30x – 15y = -15 (use the distributive property)
Second equation,
3(4x + 5y) = 3(-9)
12x + 15y = -27
(use the distributive property)
Example (cont.)
•
Combine the two resulting equations (eliminating the
variable y).
30x – 15y = -15
12x + 15y = -27
42x
= -42
x = -1
(divide both sides by 42)
Example (cont.)
•
Substitute the value for x into one of the original
equations.
6x – 3y = -3
6(-1) – 3y = -3
(replace the x value in the first equation)
-6 – 3y = -3
(simplify the left side)
-3y = -3 + 6 = 3
(add 6 to both sides and simplify)
y = -1
(divide both sides by -3)
•
Our computations have produced the point (-1,-1).
Example (cont.)
•
IMPORTANT: Check the point in the original
equations.
First equation,
6x – 3y = -3
6(-1) – 3(-1) = -3
Second equation,
4x + 5y = -9
4(-1) + 5(-1) = -9
•
true
true
The solution of the system is (-1, -1).
Use of the elimination method to combine two equations
might lead you to results like . . .
• 5 = 5 (which is always true, thus indicating that there are
infinitely many solutions, since the two equations represent
the same line), or
• 0 = 6 (which is never true, thus indicating that there are no
solutions, since the two equations represent parallel lines).
Remember, if there are infinitely many solutions, the solution
set is {(x,y) | (write in one of the two equations)}
Ch, 5 Review:
Strategy for Factoring a Polynomial:
1. Always look first to see if there is a
common factor. If so, factor out the GCF.
2. Determine the number of terms in the
polynomial and try factoring as follows:
a)
If there are two terms, can the binomial be factored by one of
the special formulas including difference of two squares, sum
of two cubes, or difference of two cubes?
b)
If there are three terms, is the trinomial a perfect square
trinomial? If the trinomial is not a perfect square trinomial, try
factoring by the British (factoring by grouping) method.
c)
If there are four or more terms, try factoring by grouping.
3. Check to see if any factors with more than
one term in the factored polynomial can be
factored further. If so, factor completely.
Example
Factor: x3 – 5x2 – 4x + 20
HOW MANY TERMS? What does that tell you?
CHECK FIRST: Is there a common factor in all terms?
ANSWER: No, but you should ALWAYS check for this first.
Solution
x3 – 5x2 – 4x + 20
= (x3 – 5x2) + (-4x + 20)
= x2(x – 5) – 4(x – 5)
= (x – 5)(x2 – 4)
= (x – 5)(x + 2)(x – 2)
Group the terms with common factors.
Factor from each group.
Factor out the common binomial factor, (x – 5).
Factor completely by factoring x2 – 4 as the difference of two
squares.
Example of factoring a trinomial
using the “British” method
Factor 8x2-10x-3
Solution:
Step 1: Find the product of a and c: 8*(-3)=-24
Step 2: Find a pair of numbers whose product is -24 and
whose sum is -10
-12*2=-24 and -12+2=-10
Step 3: Rewrite the middle term as -12x+2x, then factor by
grouping
8x2-10x-3
= 8x2-12x+2x-3
= 4x(2x-3)+(2x-3)
= (2x-3)(4x+1)
Example: Two Terms
• Factor: 81x2 – 49
Solution:
This is a difference of squares, so we can use the
formula a2 – b2 = (a + b)(a - b)
81x2 – 49 =
(9x)2 – 72 =
(9x + 7)(9x – 7).
Two terms that are cubes:
Type
Example
A3 + B3
= (A + B)(A2 – AB + B2)
x 3 + 8 = x 3 + 23
= (x + 2)( x2 – x·2 + 22)
= (x + 2)( x2 – 2x + 4)
A3 – B3
= (A – B)(A2 + AB + B2)
64x3 – 125 = (4x)3 – 53
= (4x – 5)(4x)2 + (4x)(5) + 52)
= (4x – 5)(16x2 + 20x + 25)
Solving Polynomial
Equations
Example: Solve 2x2 + 7x = 4 by factoring and
then using the zero-product principle.
Step 1 Move all terms to one side and
obtain zero on the other side. Subtract 4
from both sides and write the equation in
standard form.
2x2 + 7x - 4 = 4 - 4
2x2 + 7x - 4 = 0
Step 2
Factor.
2x2 + 7x - 4 = 0
(2x - 1)(x + 4) = 0
(Solution continued)
Steps 3 and 4 Set each factor equal to
zero and solve each resulting equation.
2x-1=0
or x + 4 = 0
2x=1
x = -4
x = 1/2
Step 5
check your solution !
Review Homework:
The homework assignment, for reviewing Unit 2
is due at the start of next class session, and it
is worth 15 points.
If you haven’t completed the Unit 1 review,
you must do that first before you can get
access to the Unit 2 assignment. Let me
know NOW if you need an extension.
Remember: You need to score at least 80% on this
homework assignment to get access to the next one.
Make sure to have the formula sheet handy as
you work on these problems.
• Any questions on the Unit 2 Review
Homework from those who have started it
already?
• If no further questions, you may open your
laptops and work on this assignment from
now till the end of class, if time remains, or
you can stay and work on it in the open
lab.
• Remember, this assignment is due at the
start of the next class session, AND you
need to score at least 80% on this to get
access to the next one.
Math TLC Open Lab Hours:
Room 203
Jarvis Hall Science Wing
Regular hours through Tuesday, May 7 (8 AM – 6:30 PM)
Finals Week Open Lab Hours:
Starting at 10 a.m. on Study Day (Wed.)
and 8 a.m. on Final Exam days
See schedule posted in the classroom
and in the open lab and online at
http://www.uwstout.edu/mathtlc

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