Download Mod 3 Ch 5-6

Mth 95 Notes
Module 3
Review
x2 – 6x – 7
`
Spring 2014
16 x 2  8 x  4
81n2 – 36
Section 5.8 Solving Equations by Factoring and Problem Solving
Remember the quadratic formula from Mth 65…
b  b2  4ac
If ax + bx + c = 0, then x =
2a
We could use the quadratic formula. Solve x2 + 5x – 6 = 0 using factoring and
solve x2 + 5x – 6 = 0.
the zero-factor property ( If a and b are
real numbers and a  b  0 , then a  0 or b  0 ).
2
5  52  4(1)(6)
x
2(1)
5  25  24
2
5  7
x
 6,1
2
x
Check by graphing in the standard window, and examine x-intercept or compute with 2nd trace (calc),
zero(2) and follow the prompts. Since these are integer results, you could also examine the table.
Why do we need both methods? Not all quadratic equations are factorable.
Solve the following quadratic equations by factoring.
Steps
1) Set equal to zero (standard form), if needed
x2 – 2x – 35 = 0
2) Factor completely, if needed.
3) Set each factor equal to zero.
4) Solve resulting equations
5) Check each solution in the original equation
(x + 2)(x – 5) = 0
2x(x + 3)(x – 1) = 0
x2 + 6x + 9 = 0
5x2 – 45 = 0
5x2 – 7x = -2
x2 + 4 = 8x – 12
Chapters 5 and 6
1
Mth 95 Notes
Module 3
Spring 2014
If fractions are included in the equation, multiply by the LCM before you factor.
x2
5
x 0
10
2
x2 x
 1  0
18 2
You can use factoring to solve higher degree equations.
x3 + 7x2 – 4x – 28 = 0
x4 – 26x2 + 25 = 0
Applications
A rectangle is 6 inches longer than it is wide. Its area is 135 square inches. Find
the dimensions of the rectangle.
Chapters 5 and 6
2
Mth 95 Notes
Module 3
Spring 2014
Remember the Pythagorean Theorem from Mth 65?
The equation a  b  c
refers to the relationship between the sides in a right triangle. The variables a and b are
the lengths of the legs and the variable c is the length of the hypotenuse, which is the
longest side of the right triangle.
2
2
2
The longer leg of a right triangle is 4 feet longer than the other leg. Find the lengths of
the two legs if the hypotenuse is 20 feet. (Hint: Use the Pythagorean Theorem)
The hypotenuse of a right triangle is 4 inches longer that the shortest leg. If the longer
leg measures 8 inches, find the other dimensions of the triangle.
Review: Solve each polynomial equation by factoring
k2 + 5k = 0
2w2 = 5w – 3
Section 6.1
Rational Functions and Multiplying and Dividing Rational Expressions
In section 5.1 we learned that MONOMIALS are terms in which the variables have
ONLY NONNEGATIVE EXPONENTS and that POLYNOMIALS are either a
monomial or a sum of monomials.
Rational Expressions result when a polynomial is divided by a NONZERO polynomial.
Examples:
Chapters 5 and 6
3
Mth 95 Notes
Module 3
Spring 2014
A rational function is given by f ( x) 
p( x)
, where p(x) and q(x) are polynomials. The
q( x)
domain of f includes all x-values such that q(x)  0, that is, all real numbers where
the ________________________________does not equal zero.
Determining the DOMAIN of a rational function:
The polynomial in the denominator CANNOT BE equal to ZERO because
3
dividing by zero is ______________________. If f(x) = , the denominator would be
x
zero when x =___. Therefore the domain of f(x) in set builder notation would be
3x3  15 x
, the domain is
7
x2  5x  6
If g ( x) 
, the domain is
x 2  25
Identify the domain of each rational function. Give your answer in set builder notation.
x2  5x  6
x6
g(x) =
h(x) =
x4
x2 1
If h( x) 
GRAPHING a rational function
2
To graph f(x) = , make a table of values and plot the points.
x
x
-4 -2 -1 -1/2 0
1/2 1
2
4
f(x)
2
does not cross
x
the line x = 0, the y-axis. A VERTICAL _________________
is a vertical line that typically occurs in the graph of a rational
function when the denominator is 0 but the numerator is not 0.
The equation for the vertical asymptote is ______________
Reducing Rational Expressions
12 4
Just as fractions can often be reduced,  , rational expressions can be reduced.
15 5
Because f(0) is undefined, the graph of f(x) =
x  2x 1
12 4
12 4  3 4

 because 
 . So, 2
x  6x  7
15 5
15 5  3 5
2
Remember you can only cancel factors, not terms!
Chapters 5 and 6
x3
 3.
x
4
Mth 95 Notes
Module 3
Spring 2014
Simplify each rational expression. Factor and cancel (remove the hidden factors of 1)
y  2
4y  8
x2  4
x2
15 x
5 x3
x 2  7 x  18
x 2  3x  2
x3  4 x 2  3x  12
x4
Review
Factor x3  3x 2  9 x  27 completely.
Give the domain of
Solve 2m2  7m  3 by factoring.
x2  7 x  6
x2  2 x
Simplify
x 2  x  20
.
3 x  12
5x
x 5
Multiplying Rational Expressions
1 4 1 2  2 2

Remember fractions:  
2 3
23
3
Steps in multiplying rational expressions
1. Factor everything
5 x3 2 y 4

4 y 15 x
x 2 1 x  2

x 2  2x x 1
2. Cancel, if you can
3. Write down what’s left
Multiply the following rational expressions.
x  3 x2  2x  8

x  4 x2  6 x  9
Chapters 5 and 6
5
Mth 95 Notes
x 2  25
x2

2
x  3x  10
x
Module 3
Spring 2014
x 2  6 x  9 4 x  12

2 x 2  18 5 x  15
Dividing Rational Expressions
2 x3 8 x
3 1 3 2
3
2 3

Remember fractions:    
 
4 2 4 1 22 1 2
5 y 6 y2
Steps in dividing rational expressions
1. Flip the rational expression following the division sign (that gives you the
reciprocal of the divisor) and replace the division symbol with a multiplication
symbol.
2. Completely factor everything
3. Cancel what you can
4. Write down what’s left
x 2  3x  10 x 2  5 x  6
 2
2x
x  3x
10n  15 6n  9

n2  1
n 1
4
40

y  6 42  7 y
x 2  x  12 x 2  7 x  12

2 x2  9 x  5 2 x2  7 x  4
Simplify
x 2  x  20 x 2  16
3


3x  15
x 4 x 5
Chapters 5 and 6
2 x 2  12 x x 2  3x  18

5
15 x
6
Mth 95 Notes
Module 3
Spring 2014
Review
Simplify
x 2  x  20
.
3 x  12
Give the domain in set builder notation.
16w2  4w
3w2  14w  8
6.2 Additions and Subtraction of Rational Expressions
Remember to add or subtract fractions, the fractions must have a common denominator (
the LCM of the denominators).
3 2 5
1 5 1 3 5 2 3 10 13
       
Ex1  
Ex2
7 7 7
4 6 4 3 6 2 12 12 12
Steps in adding and subtracting rational expressions that have a common denominator.
1. Add or subtract the numerators
2. Factor everything, if you can
3. Simplify, if you can
2 x 5x

3y 3y
5y
5

y 1 y 1
x 2  x  4 3x  4
 2
x2  4
x 4
y  3 5y  2

2y
2y
Steps in adding and subtracting rational expressions that have unlike polynomial
denominators.
1. Factor each polynomial in the denominator completely
2. List each factor the greatest number of times it occurs in each denominator.
3. The product of these factors is the LCD.
4. Rewrite each expression with the LCD by multiplying each fraction by one.
5. Add or subtract the numerators
6. Factor everything, if you can
7. Simplify, if you can
Chapters 5 and 6
7
Mth 95 Notes
3
5
 3
2
7x
2x
Module 3
LCD =
x 1
5

x
x 5
LCD =
Spring 2014
4
3

5 x 4 xy
LCD =
x
x 1

2x  4
2x
LCD =
1
1

2
2
x  3x  2
x  x 2
x2
3

x 2  9 x 2  4x  3
x2  6
x4

2
x  9 x  18 x  6
x
x 1
 2
x  3 x  2x  3
Chapters 5 and 6
8