Integrated AlgebraRegents Review #1

Rational ExpressionsScientific Notation

Trigonometry

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

Examples:

34x3

5xx2 1

56xx168xx

2

2

Rational ExpressionsSimplifying Rational Expressions

with Monomial Numerators and Denominators

3

3

24xyy4x

6y1x 22

2

2

6yx

Method 1: Use laws of exponents

Method 2: Expand and divide out common factors

3

3

24xyy4x

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

(24)(x)(y)(y)(y) 2

2

6yx

Rational ExpressionsSimplifying Rational Expressions

with Polynomial Numerators and Denominators

A)

1)2)(x(x2)x(x

23xx2xx

2

2

• 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.

B)

5)(3)(x)(x5)x(2)(x)(x)(

5)3x(x5)(x2x

15x3x10x2x 2

2

23

xx + 1

32x

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

82xxx

xx20xx

22

2

2)4)(x(xx

1)x(x4)5)(x(x

2)1)(x(x5)(x

Factor all the numerators and denominators!

Rational ExpressionsWhen dividing rational expressions…1)Keep, Change, Flip2)Factor all the numerators and denominators3)Divide out common factors

6xx9x

65xx9x3x

2

2

2

2

9x6xx

65xx9x3x

2

2

2

2

3)3)(x-(x2)3)(x-(x

3)2)(x(x3)3x(x

Keep the first fractionChange division to multiplicationFlip the second fraction (reciprocal)

3x3x

Rational Expressions1) When adding and subtracting rational

expressions, find a common denominator if necessary.

2) Create equivalent fractions using the common denominator(Multiply by FOOs).

3) Add or subtract numerators and keep the denominator the same.

4) Simplify your final answer if possible.

FORM OF ONEEx:

xx

or22

Rational ExpressionsWhat is the sum of and expressed in simplest form?

5y2y 5y

10

5y102y

(4)5y

12y(3)2(2)1(1)

5y102y

5y10

5y2y

5y5)2(y

5y102y

Add numerators and keep the denominator.

Simplify by factoring the numerator and denominator.

2

Rational Expressions

23x2x

9x4

LCD (Least Common Denominator): 9x2

22 9x?

9x?

xx

33

22 9x2)3(x

9x4x

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

29x6x

2222 9x6-3x-4x

9x6)(3x-4x

9x63x

9x4x

FOO FOO

Rational Expressions

21

x6

8x4

41

8x4

8x2x

11

8x4

41

2x2x

8x42x

2xx

2x12

xx

21

x6

22

2x12x

2x12x

8x42x

8x(x + 12) = 2x(2x + 4) 8x2 + 96x = 4x2 + 8x 4x2 + 88x = 04x(x + 11) = 0 4x = 0 x + 11 = 0 x = 0 x = -11

FOO

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.

Reject 0 because it makes the equation undefined.

Solution: x = -11

FOO

Scientific Notation

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

Representing Numbers in Scientific Notation

Scientific NotationMultiplying Numbers in Scientific NotationUse the commutative property and laws of exponents

3101.28

41

4

95

10101.28

1012.8

10103.24

95 102.3104 Calculator Corner:

1)Press MODE2)Select SCI (see top row) , ENTER3)Press 2nd MODE to return to home screen4)Enter expression into the calculator (use the expression on this slide to practice) (4x10^5)(3.2x10^-9)5) Press ENTER6) The expression 1.28E-3 means 1.28 x 10-3

7)Go to MODE and select NORMAL to exit scientific notation

Scientific Notation

18101.4

6

12

6

12

1010

45.6

104105.6

Dividing Numbers in Scientific NotationCalculator Corner:

1)Press MODE2)Select SCI (see top row) , ENTER3)Press 2nd MODE to return to home screen4)Enter expression into the calculator (use the expression on this slide to practice)5)Put the numerator and denominator in ( ) (5.6x10^-12)/(4x10^6)6) Press ENTER7) The expression 1.4E-18 means 1.4 x 10-18

8)Go to MODE and select NORMAL to exit scientific notation

TrigonometryTrigonometric Ratios

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

34

(4)54

(3)43

(2)53

(1)

Answer: (1)

53

106

sin

hypopp

θsin

TrigonometryFinding Sides of a Right Triangle•Use the Pythagorean Theorem when given two sides

•Use Trigonometry when given a side and an angle

a2 + b2 = c2

32 + 42 = c2

9 + 16 = c2

25 = c2

5 = c c5

c4.984...

c37sin

337)c(sin3c3

137sin

c3

37sin

hypopp

Asin

Calculator must be in degrees(See MODE)

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

Pythagorean Theorem SOH CAH TOA

TrigonometryFinding Angles of a Right Triangle > Use inverse trig ratios

1

1

1

osin m θ

h

acos m θ

h

otan m θ

a

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

1

1

otan m θ

a

24tan 53.1301...

18οthe angle measures 53

Calculator:2nd TAN

Trigonometry

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

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.

1

1

acos m θ

h

35cos m θ

50

measure oft he angle 45.572996ο46

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

Looking for this angle

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!