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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