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    Intermediate Algebra
    Spring Semester (S2) Notes
     
    Chapter 6 - "Rational Expressions"
     rational wordle
    Factor quadratic trinomials using diamond problems.
    Then cancel common binomial factors.
     
    Worksheet--What Do You Call an Alligator That Sneaks Up and Bites You From Behind? 
     #1-5
    PZ107 1-5  
     Also, use the difference of two squares formula:  (a-b)(a+b)
     #6-8
     PZ107 6-8
     Find greatest common factors (GCF).
    #9-10
    PZ107 9-10
     
    #11-13
    PZ107 11-13
     
    #14-15
    For #14, multiply the leading coefficient by the constant term,
    then do a diamond problem with -4 and +3.  Factor by grouping (or box method.)
    PZ107 14-15  
     ATAILGATOR
     
    What do you do when you have 3-x instead of x-3?  Factor out a "-1".
    (3-x) = -1(x-3).  This means that (3-x) / (x-3) reduces to -1.
     Is that true?
     
    (3-5) / (5-3) = -2 / 2 = -1 and
    (7-4) / (4-7) = 3/(-3) = ?.
    This is true for all numbers.
     
    PZ108 "Cryptic Quiz"
    1. What do you call a skydiver with the flu?
    PZ108 1-7
     (9-x^2) = (3-x)(3+x)
    -(x^2-9) = -1 (x-3)(x+3)
     
     2. How do you crash a houseboat party?
     
    PZ108 8-14
     
     
     

     
    Simplifying Rational Expressions
     
    PZ109 "What do you call an insect that plays drums?"
    PZ109 ITH
    PZ109 KH
    PZ109 KM  
    PZ109 CI  
    A RHYTHMIC TICK
     
    BOOKS NEVER WRITTEN
    PZ110 1-6  
     PZ110 7-12
     
     PZ110 Books
     
     
    Multiplying Rational Expressions
     
     
    What Do You Call a Message Printed on a Lion With Chickenpox?
     PZ111 2 8
    PZ111 1 3 5  
    PZ111 4
     
    PZ111 9 7  
     PZ112 Why Are Ancient Stories Like Feet?
     PZ112
    (Match the work below with its problem and answer.)
    PZ112 2, 6, 8, 3  
    PZ112 9, 7, 4
    PZ112 7,4,1,5
     
    THEY ARE BOTH LEG ENDS

    Dividing Rational Expressions
     
    To divide rational expressions, flip the second fraction to multiply by the reciprocal.
     p373 51
    Then factor and cancel common factors.
    p373 52
    Note the difference of squares a^2 - b^2 = (a - b)(a + b).
    p373 53
    (There is a mistake in the photo.  The corrected solution is below.) 
    Number 53 solution  
    Be careful, common factors in the numerator do not cancel.
    p373 54 & 57-58  
    The next problem has a difference of cubes factorization:
    a^3 - b^3 = (a - b)(a^2 + ab + b^2)
    p373 62-63
     #67 involves factor by grouping:
    ab - 2b +3a - 6 =
    b(a - 2) + 3(a - 2) =
    (a - 2) (b + 3)
    p373 64-68  
    The denominator in #68 involves factor by grouping.
     

    ADDING RATIONAL EXPRESSIONS
     
     Common Denominators for PZ119
    What Lives in the Sea and Yells?
     PZ118
     PZ119 1-5
     PZ118 6-11
    A CLAM SHOUTER 
     A CLAM SHOUTER
     
    Why Did Orgo Take a Bath After Walking Through Mudsucker Swamp?
     Find common monomial denominators.
    PZ119 hints 1-5
    IT WAS A DIRTY TREK
    IT WAS A LONG TREK  
    The numerators are:
    #6 5a(a+4) + 3(2a-1) =
    #7 3(a+6) + a(4a+3) =
    #8 3a^2(a-4) +6a(1) + 2(7-3a) =
    #9 a(3a+b) + b(5a-2b) =
    #10 b(2a+2) + 7(b-9) =
     
    Finding common polynomial denominators.
     The work to find each common denominator is shown below:
     
    Match the work with its problem and answer on this worksheet.
     PZ120
     
    Match the work in the photo gallery above with each problem and answer below.
    PZ121 Snowman  
     snowman 1-5 snowman 6-10
     

    Complex Fractions
     fractions
     
    multiply by denominator's reciprocal  
     
    Fractions imply division.
    complex fraction examples  
    You can also multiply top and bottom by the same quantity to remove denominators.
    simplify x squareds  
     
     p389#1
    p389 #1  
     To divide a fraction, multiply by its reciprocal.
     
     p389#3
    p389#3  
    Find common denominators in the numerator and denominator.
     
    p389#5
    p389#5
    Cancel like algebraic expressions.
     
    p.389#7 
    p389 seven
    Find common denominators, the reciprocal, then distribute.  
    Why can't 9x^2 terms cancel?
     
    p389 #9
    p389 nine
    Find common denominator, then multiply by reciprocal.
     
    p389 #11
    p389#11
    Multiply numerator by denominator's reciprocal.  Can the 2x's cancel?
     
    p389 #13
    p389#13
    Multiplying top and bottom by x^2 cancels each denominator.
     
     
    p389 #15  
    p389#15  
    Note the difference of cubes factor.
     

    Polynomial Long Division
    dividend over divisor
     
    division vocabulary
     
    divide     Rewrite this expression with the long division symbol:
     
    polynomial division  
    answer  
     
     
    2x
     
    2x(x+5)
     
    Here's an animated gif on how to complete the problem.
     
    long division steps  
    Here is each step of the problem:
     
    1) Setup the problem.      2)  What is 2x^2/x?      3)  Multiply 2x(x)      4)  Multiply 2x(5).
    step 0    2x^2/x = 2x       multiply 2x by x   multiply 2x by 5
    5) 2x(x+5) = ?      6)  Subtract (2x^2+10x)      7) by reversing signs,      8) add to get -3x.
    step4    subtract   ...or reverse signs...   ...and add to get -3x 
    9) Bring down -15.      10) -3x/x = -3      11) multiply -3(x)      12) and -3(5) 
    bring -15 down    -3x/x = -3   multiply -3 by x   multiply -3(5)
    13) write -3(x+5) product.      14) Subtract -3x-15,      15) or add 3x+15      16) to get 0.
    -3(x+5)=   subtract (-3x-15)   or add opposites   step16  
    The remainder is zero.
     
    Remainder 0 
    Here's the final solution.
     
     
    2x-3
     
     
     polynomial long division1
    Polynomial Long Division2  
     
    What if there is a remainder?
     
    2x+1 + 2/(x-3)  
     
     
    Divide
    Problem21  
     

    Number21

     

     

     
     

    There is a remainder of 5. 

    So the final answer is: number21 answer
     
     

     

    Here's how to do #23 step by step:
     
    number23                     number 23
     
     

    Synthetic Division
    Divide this quartic by this binomial.
    setup division  
    Account for 0 x-squareds when writing coefficients.
    how to synthetically divide
     

    Now divide  syn div ex2   using synthetic division:
    Solve x - 4 = 0 first to get the divisor's root.
    Then write the coefficients in descending order.
     
    syn div example
     
     
    The remainder of 304 is written as a fraction: syn div ex2 ans .
     
    This next example is illustrated with a gif:
     
    Syn Div setup  
    Here's how to proceed:
    synthetic division gif
     
    The remainder is 19: 
     syn div answer The final answer is:
    Syn Div answer  
     
     Syn Div Solution
     
     

     
     

     

    SOLVING RATIONAL EQUATIONS
     
     PZ AB PZ131 CD
     PZ131 E
    PZ 131 E
     
    PZ 131 F
     
     
     
     pz131 G
    pz131 G  PZ131 G
     
     
     

     
    Rational Exponents
     
     
    rational radical rule
     
     radicand vocabulary fractional exponents
    8 to the 1/3  
    example
    examples

     perfect powers
     

     
     
    Combining Like Radicals
     
     
    equation 1
    are like radicals because they have the same index (root number which is 3) and the same radicand (number under the radical which is 5.

    equation 2
    are not like radicals because they have different radicands 8 and 9.

    equation 3
    are like radicals because they have the same index (2 for square root) and the same radicand 2 x.

    Add and Subtract Like Radicals

    Only like radicals may be added or subtracted.
    Examples


    Simplify the following expressions

    equation 4

    Solutions to Above Examples


    The above expressions are simplified by first factoring out the like radicals and then adding/subtracting. equation 5
    More Examples


    Simplify the following expressions

    equation 6

    Solutions to Above Examples


    The above expressions are simplified by first transforming the unlike radicals to like radicals and then adding/subtracting

    equation 7


    equation 8



    equation 9

    When it is not obvious to obtain a common radicand from 2 different radicands, decompose them into prime numbers. Decompose 12 and 108 into prime factors as follows.

    equation 10

    We now substitute 12 and 108 by their prime factors and simplify

    equation 11


    equation 12


    equation 13


    Questions With Solutions

    Simplify the following expressions
    equation 14 Solutions
    equation 15
    3. The 3 radicands in the given expression -√ 32 - 2√ 50 + 3√ 200 are different but note that 32, 50 and 200 may be written as 2 times a number that is a perfect square as follows: 32=2 * 16, 50=2 * 25 and 100=2 * 100. Substitute in the given expression and simplify.

    equation 16
    equation 17
    equation 18

    Decompose 28 and 63 into prime factors as follows: 28=2 2 * 7 , 63=3 2 * 7 and substitute into the given expression and simplify

    equation 19
    equation 20
    7.
    equation 21
    equation 22 
     
    number sets
    i rotation  
    i powers
    i powers
    i examples
    Nine Zulu Queens Rule China  
    be rational, get real  keepin it real
     

    Completing the ...  
    completeing square
    SQUARE! 
     
    completing tiles  
     
    x^2 + 4x + ?
     
     
    complete with 4 add 2 more  
     
     x^2 + 6x + ?
    add 9 three squared  
     
    (x+4)^2 =  
    half of 8 squared
     
     
    Click on pic below for active demonstration:
     
    complete 10x  
    complete process
    (half of b) squared  
     
     
     Solve x^2 + 6x + 8 = 0
    complete the process  

     
     complete the example
    complete irrational          complete real
    complete the real
    complete the example
    complete the math

     THE QUADRATIC FORMULA
     quadratic formula
    quadratic coefficients  
     
    Click here for animated slide show
    with sound effects (lasers, whips, and explosions!)
     
    Deriving the formula
     
    applying quadratic formula

     solutions to x^2 + 6x + 3 = 0
     
    QF example
     
     Use the formula to solve: qf ex1
     
    discriminant - rational
     
     Use the formula to solve: qf ex2  
     
    dscriminant - irrational
     
     
     Use the formula to solve:  qf ex3  
    discriminant - imaginary  
     
    how to use formula  
     
     When am I ever gonna use this?
     

     
    GRAPHING QUADRATIC EQUATIONS
    vertex = -b / (2a)
     
     
    Window for #45                                                   Window for #50
    p575 45 window p575 50 window  
    Two numbers add to 11:  x + y = 11.   Their product xy is 11.  
    Solve for y:  y = 11 - x.   Then product is x(11-x). 
    Graph and find vertex for maximum product.
    -b/2a = -11/(2*-1) = 5.5
     

     
    PZ 223 Calculator Keystroke Screens
    Click PZ223 Calculator for the rest of the keystroke solutions.
     
     
     
    parabola shift
     

    Composition of Functions
    (fog and gof)
     
    fog odd
     
    function machines
    function cube
    function boxes  
     
    apples to oranges
     
    output is input
     
    step by step
     
    gof example
     
    sum and difference  
    Fill in the table below:
    function table  
     
     
     
Last Modified on February 9, 2018