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.

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).• Unit 2 review homework (Chapters 4, 5, 8):

Due today.

• 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 next class session.

• 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. Check with your teacher for due date and time.

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 last day of classes (next Tuesday, May 10):

Monday–Thursday: 8:00 a.m. - 6:30 p.m.

Study Day and finals week:

Wednesday, 5/11: 10:00 a.m. - 4:00 p.m.Thursday, 5/12: 8:00 a.m. - 6:00 p.m.Friday, 5/13: 8:00 a.m. - 6:00 p.m.Monday, 5/16: 8:00 a.m. - 6:00 p.m.Tuesday, 5/17: 8:00 a.m. - 2:00 p.m.

Now please

CLOSE

YOUR LAPTOPSand turn off and put away your cell phones.

Any questions on the Unit 2 Review Homework?

Final Exam Review, Unit 3

Unit 3 Covers:

Chapter 6 (Rational expressions)

Don’t forget to review the application problems from section 6.6 homework and your work on the offline worksheet.

(Remember those fun problems about planes and cars, and three people doing a job at different rates?)

These were a common source of trouble for many students in previous semesters, so make sure you study the application problems on the Practice Test, and then review the section 6.6 homework if you need more work on those problems.

Unit 3:Chapter 6 (Rational Expressions)

IMPORTANT:Almost all problems in Chapter 6 on rational

expressions require you to use one or more techniques for factoring polynomials.

DON’T FORGET TO REVIEW THESE FACTORING METHODS AND KNOW HOW AND

WHEN TO USE EACH ONE!

Two methods that students often have trouble with are factoring by grouping and sums/differences of cubes, so make sure you review problems using these methods.

Domain example:Find the domain of the expression

20

432

2

xx

xx

Solution: Set the denominator equal to zero and solve for x. (Notice that this boils down to a factoring problem.)

x2 – x – 20 = 0(x – 5)(x + 4) = 0(x – 5) = 0 gives x = 5, and (x + 4) = 0 gives x = -4

Therefore the domain is {x | x ≠ 5 and x ≠ -4}

Multiply the following rational expressions.

3 2 2

3 3 3

6

2 2

a a b a b a

a a a b

Example

2 2

3:

aANSWER

a ab b

Solution: Factor each polynomial completely,then cancel all factors that appear on both the top and the bottom.

NOTE: This problem uses factoring by grouping, factoring

out a GCF (twice!), and the sum of cubes formula.

Whew!

Add the following rational expressions.

65,

6

422 xx

x

xx

656

422 xx

x

xx

)2)(3()2)(3(

4

xx

x

xx

)3)(2)(3(

)3(

)3)(2)(3(

)3(4

xxx

xx

xxx

x

)3)(3)(2(

3124 2

xxx

xxx

)3)(3)(2(

122

xxx

xx

Example

Dividing a polynomial by a monomial:Divide each term of the polynomial separately by the

monomial.

a

aa

3

153612 3 aa

a

a

a

3

15

3

36

3

12 3

aa

5124 2

Example

15232837 2 xxx

x4

xx 1228 2 35 x

5

1535 x

Divide 7x into 28x2.

Multiply 4x times 7x+3.

Subtract 28x2 + 12x from 28x2 – 23x.

Bring down -15.

Divide 7x into –35x.

Multiply -5 times 7x+3.

Subtract –35x–15 from –35x–15.

Nothing to bring down.

15

So our answer is 4x – 5.

Example Dividing a trinomial by a binomial:

HOW WOULD YOU CHECK THIS ANSWER???

Example of a complex fraction:

10 3x . 5 6x

Solution: view as a division problem:

10 ÷ 5 = 10 · 6x = 4 3x 6x 3x 5

Example: Solving a rational equation:

Solve 2x + 1 = 1 . 2x – 1 x 2x – 1

Solution: 1. Multiply both sides by the LCD x(2x – 1). This gives 2x(x) + 1(2x – 1) = 1(x)2. Now simplify this equation: 2x2 + 2x – 1 = x 2x2 + x – 1 = 03. Solve either by factoring or by using the quadratic formula. Answers: x = -1, x = 1/2

Example (continued):

NOW CHECK BOTH ANSWERS in original equation:

Note that when you plug in x = 1, everything checks out OK ( you get -1/3 = -1/3)

BUT when you plug in x = 1/2, you get zeroes in some denominators, so this number is NOT a solution. (it gives 1 + 1 = 1

0 ½ 0

So the final answer is just x = -1

(if you write x = -1, 1/2, you’ll get ZERO credit.)

REMINDER: Comprehensive Practice Final:

Once you finish the Unit 3 and 4 Final Review homework assignments 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 last day of classes (next Tuesday, May 10):

Monday–Thursday: 8:00 a.m. - 6:30 p.m.

Study Day and finals week:

Wednesday, 5/11: 10:00 a.m. - 4:00 p.m.Thursday, 5/12: 8:00 a.m. - 6:00 p.m.Friday, 5/13: 8:00 a.m. - 6:00 p.m.Monday, 5/16: 8:00 a.m. - 6:00 p.m.Tuesday, 5/17: 8:00 a.m. - 2:00 p.m.