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? 
    PZ107 1-5  
     Also, use the difference of two squares formula:  (a-b)(a+b)
     PZ107 6-8
     Find greatest common factors (GCF).
    PZ107 9-10
    PZ107 11-13
    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  
    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  
    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?
    (Match the work below with its problem and answer.)
    PZ112 2, 6, 8, 3  
    PZ112 9, 7, 4
    PZ112 7,4,1,5

    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.

     Common Denominators for PZ119
    What Lives in the Sea and Yells?
     PZ119 1-5
     PZ118 6-11
    Why Did Orgo Take a Bath After Walking Through Mudsucker Swamp?
     Find common monomial denominators.
    PZ119 hints 1-5
    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.
    Match the work in the photo gallery above with each problem and answer below.
    PZ121 Snowman  
     snowman 1-5 snowman 6-10

    Complex 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  
     To divide a fraction, multiply by its reciprocal.
    Find common denominators in the numerator and denominator.
    Cancel like algebraic expressions.
    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
    Multiply numerator by denominator's reciprocal.  Can the 2x's cancel?
    p389 #13
    Multiplying top and bottom by x^2 cancels each denominator.
    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  
    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.
     polynomial long division1
    Polynomial Long Division2  
    What if there is a remainder?
    2x+1 + 2/(x-3)  





    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



     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  

     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.

    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
    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
    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

     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?

    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