Slide 1

Integrated Algebra
Regents Review #1
Rational Expressions
Scientific Notation
Trigonometry


Slide 2

Rational Expressions
Rational Expressions are fractions (ratios) that
contain polynomial expressions in the numerator
and denominator.
Examples:

4x

3

2

x 1
5x

3
2

x  8x  16
2

x  6x  5


Slide 3

Rational Expressions
Simplifying Rational Expressions
with Monomial Numerators and Denominators
Method 1: Use laws of exponents
3

4x y
24xy

3



2

1x y
6

2



x

2

6y

2

Method 2: Expand and divide out common factors
3

4x y
24xy

3



(4)(x)(x)(x)(y)
(24)(x)(y)(y)(y)



x

2

6y

2


Slide 4

Rational Expressions
Simplifying Rational Expressions
with Polynomial Numerators and Denominators

• Factor the numerator and denominator
• Divide out common factors
Remember: Monomials can only be simplified with monomials and
binomials can only be simplified with binomials that are exactly the same.

x  2x
2

A) x
B)

 3x  2

2

2x

3

3x

2

 10x
 15x



2

x(x  2)
(x  2)(x  1)

2x (x  5)
2



3x(x  5)





x
x+1

(2)(x)(x)( x  5)
(3)(x)(x  5)



2x
3


Slide 5

Rational Expressions
When multiplying rational expressions, factor all the
numerators and denominators. Divide out common factors.

x  x  20
2

x x
2

(x  5)(x  4)
x(x  1)





x
x  2x  8
2

x
(x  4)(x  2)

(x  5)
(x  1)(x  2)

Factor all the numerators and
denominators!


Slide 6

Rational Expressions
When dividing rational expressions…
1) Keep, Change, Flip
2) Factor all the numerators and denominators
3) Divide out common factors

3x  9x
2

x  5x  6
2

3x  9x

2



2

x  5x  6
2

3x(x  3)
(x  2)(x  3)

x 9
x x6
2

x x6
2





x 9
2

Keep the first fraction
Change division to multiplication
Flip the second fraction (reciprocal)

(x - 3)(x  2)
(x - 3)(x  3)



3x
x3


Slide 7

Rational Expressions
1) When adding and subtracting rational
expressions, find a common denominator if
necessary.
2) Create equivalent fractions using the common
OF ONE
denominator(Multiply by FOOs). FORM
x
Ex: 2
2

or

x

3) Add or subtract numerators and keep the
denominator the same.
4) Simplify your final answer if possible.


Slide 8

Rational Expressions
What is the sum of

(1)

2y
y5

1



(2)

10
y5

2y

and

y5

2



(3)

y5

12y
y5

2y  10
y5

Simplify by factoring the
numerator and denominator.

10

expressed in simplest form?

(4)

2y  10
y5

Add numerators and
keep the denominator.

2y  10
y5



2(y  5)
y5



2


Slide 9

Rational Expressions
FOO

x
x



4
9x
?
2

9x
4x

2

9x
4x
9x

2






x2
3x
?
9x

2

FOO



3
3

2

3(x  2)
9x

3x  6
9x

2

LCD (Least Common Denominator): 9x2

2



Find equivalent fractions with a common
denominator by multiply by a FOO (form of one).

4x - (3x  6)
9x

2



4x - 3x - 6
9x

2



x6
9x

2


Slide 10

Rational Expressions
When solving rational equations (equations with algebraic fractions), combine
fractions and set up a proportion. Remember: A common denominator is
needed to add or subtract fractions.

1
4
2x
2x
FOO




1
4

2x
8x




4
8x
4
8x





6
x

1

2

1

2




6
x

4

12

8x

2x

2x  4

1

8x

2



1
2



x
2x

2x  4

x  12

8x

2x

x
x
FOO



x  12
2x

8x(x + 12) = 2x(2x + 4)
8x2 + 96x = 4x2 + 8x
4x2 + 88x = 0
4x(x + 11) = 0
4x = 0 x + 11 = 0
x=0
x = -11
Reject 0 because it makes the
equation undefined.

Solution: x = -11


Slide 11

Scientific Notation
Representing Numbers in Scientific Notation
If this a skill you
have not
mastered and
need additional
instruction,
re-watch
FLIP #3.


Slide 12

Scientific Notation
Multiplying Numbers in Scientific Notation
Use the commutative property and laws of exponents

4  10 3 . 2  10 
4  3.2 10  10 
9

5

9

5

12.8  10

4

1.28  10  10
1

1.28  10

3

4

Calculator Corner:
1)
2)
3)
4)

Press MODE
Select SCI (see top row) , ENTER
Press 2nd MODE to return to home screen
Enter expression into the calculator (use the
expression on this slide to practice)
(4x10^5)(3.2x10^-9)
5) Press ENTER
6) The expression 1.28E-3 means 1.28 x 10-3
7) Go to MODE and select NORMAL to exit
scientific notation


Slide 13

Scientific Notation
Dividing Numbers in Scientific Notation
 12

5.6  10
4  10
5.6
4



6

10

 12

10

1.4  10

6

 18

Calculator Corner:
1)
2)
3)
4)
5)
6)
7)
8)

Press MODE
Select SCI (see top row) , ENTER
Press 2nd MODE to return to home screen
Enter expression into the calculator (use the
expression on this slide to practice)
Put the numerator and denominator in ( )
(5.6x10^-12)/(4x10^6)
Press ENTER
The expression 1.4E-18 means 1.4 x 10-18
Go to MODE and select NORMAL to exit scientific
notation


Slide 14

Trigonometry
Trigonometric Ratios

What ratio represents the sine of the
indicated angle pictured to the right?
(1)

3
5

(2)

3
4

(3)

4
5

(4)

Answer: (1)
sin θ 

4
3

sin



opp
hyp
6

10



3
5


Slide 15

Trigonometry
Finding Sides of a Right Triangle
• Use the Pythagorean Theorem when given two sides
• Use Trigonometry when given a side and an angle
Pythagorean Theorem

SOH CAH TOA

sin 37 
sin 37
1



3
c
3
c

3  c(sin 37)
a2

b2

c2

+ =
32 + 42 = c2
9 + 16 = c2
25 = c2
5=c

sin A 

opp

3

hyp

sin 37

Calculator must be in degrees
(See MODE)

c

4.984...  c
5c

Substitute
known
values into
the trig
ratio. Solve
for the
variable by
cross
multiplying.


Slide 16

Trigonometry
Finding Angles of a Right Triangle

> Use inverse trig ratios

Find the measure of the indicated angle to
the nearest degree.

sin

1

co s

ta n

1

1

o
   mθ
h
a
   mθ
h
o
   mθ
a

tan

1

o
   mθ
a

Calculator:
24


1
2nd TAN
tan 

53.1301...

 18 

the angle m easures  53

ο


Slide 17

Trigonometry
A 50 ft. ladder leans against a building. The foot of the ladder is 35 feet from
the building. To the nearest degree, find the measure of the angle that the
ladder makes with the ground.

• Are you looking for an angle or side?
• Trig Ratio or Inverse Trig Ratio?
• Draw a picture of the situation
co s

co s

1

1

a
   mθ
h
 35 

  mθ
 50 

Looking for this angle

m e a su re o f th e a n gle  45.572996
 46

ο

Remember: Calculator must be in degrees in order to do a
trigonometry problem. Go to Mode and highlight degree (ENTER)


Slide 18

Now it’s your turn to review on your own!
Use the information presented today to help you practice
questions from the Regents Exams in the Green Book.
See halgebra.org for the answer keys.

Integrated Algebra Regents Review #2
Monday, June 16th
BE THERE!