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Math Help - help with polynomials

  1. #1
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    help with polynomials

    Im stuck on the following questions and need some help. Sorry I dont know how to do X squared on the keyboard so X^2 means X squared and etc.

    1. Simplify 4X^5 + 16X / 4X
    b. Simplify 2X^2 +11X +12 / 2X^2 + X-3

    2. Divide X^3 - 5X^2 + X + 10 by X-2
    b. Divide X^3 - 27 by X-3

    3. (X-2) is a factor of f(X) = X^3 + 7X^2 + qX - 40 therefore find q

    4. Show that (2X-1) is a factor of 2X^3 + X^2 + X-1 and then find the quadratic factor.

    5. f(X) = X^3 + pX^2 -49X -120 given that f(-5) = 0, find p
    b. Factorise f(X) completely into linear factors

    6. A function is given as f(X) = aX^3 + bX^2 + X-10. When f(X) is divided by (X-2) the remainder is 36. (X+2) is a factor of f(X), show that a=2 and then find the value for b.

    7. A function is given as f(X) = X^3 + aX^2 + bX + 6. Find in terms of a and b, the remainder when f(X) is divided by

    a. (X-2)
    b. (X+3)

    then given that these remainders are equal express a in terms of b.

    8. Find the remainder when 2X^2 + 4X - 5 is divided by (2X+1)

    Sorry I asked for so much, these are past paper questions that I don't understand, so sorry to cause trouble and if you dont mind can you also write out your workings as well to see how to come to that conclusion so I can handle questions of the smilar sort as future reference.

    This needs to be in by tomorrow so its in a hurry.

    Thank You.
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  2. #2
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    Quote Originally Posted by novadragon849 View Post
    ...

    1. Simplify 4X^5 + 16X / 4X
    b. Simplify 2X^2 +11X +12 / 2X^2 + X-3

    ...

    Hi,

    I assume that you mean:

    (4x^5 + 16x)/(4x). Factor the numerator and cancel:

    (4x(x^4 + 4))/(4x) = x^4 + 4

    to b)
    I assume that you mean:

    (2x^2 +11x +12) / (2x^2 + x - 3)

    Factor numerator and denominator:

    ((x+4)(2x+3))/((x-1)(2x+3)) = (x+4)/(x-1)

    EB


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  3. #3
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    Quote Originally Posted by novadragon849 View Post
    ...
    2. Divide X^3 - 5X^2 + X + 10 by X-2
    b. Divide X^3 - 27 by X-3
    ...

    Hi,

    Code:
    (x^3 - 5x^2 + x + 10)  (x-2) = x^2 - 3x - 5
     x^3 - 2x^2
    ------------
           -3x^2 + x
           -3x^2 + 6x
           -----------
                 -5x + 10
                 -5x + 10
                 ---------
                        0
    Code:
    (x^3 - 27)  (x-3) = x^2 + 3x  + 9
     x^3 - 3x^2
     -----------
           3x^2 - 27
           3x^2 - 9x
           ----------
                  9x - 27
                  9x - 27
                  --------
                        0
    EB
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  4. #4
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    Quote Originally Posted by novadragon849 View Post
    ...
    3. (X-2) is a factor of f(X) = X^3 + 7X^2 + qX - 40 therefore find q
    ...

    Hi,

    do the long division as usual. The remainder must be zero:
    Code:
    (x^3 + 7x^2 + qx - 40)  (x-2) = x^2+9x+(q+18)
     x^3 - 2x^2
     ---------
            9x^2 + qx
            9x^2 - 18x
            -------
                 (q+18)x -40
                 (q+18)x - 2q-36
                ----------------
                           2q - 4
    This remainder must be zero: 
    
    2q - 4 = 0  ===>  q = 2
    Do #4 the same way as #3.
    EB
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  5. #5
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    Quote Originally Posted by novadragon849 View Post
    ...
    5. f(X) = X^3 + pX^2 -49X -120 given that f(-5) = 0, find p
    b. Factorise f(X) completely into linear factors
    ...

    Hi,

    first plug (-5):

    f(-5) = -125 + 25p + 245 - 120 = 0
    25p = 0 ==> p = 0

    Therefore:

    f(x) = x^3 - 49x -120

    Split 49 into 24 + 25:

    f(x) = x^3 - 25x - 24x - 120
    f(x) = x(x^2 - 25) - 24(x+5)
    f(x) = x(x+5)(x-5) - 24(x+5)
    f(x) = (x+5)(x^2 + 5x -24)
    f(x) = (x+5)(x+8)(x-3)

    EB


    It's very late now. I have to catch some sleep. Sorry that I cann't help you further on.
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  6. #6
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    wow earboth your amazing. Anyway I was wondering if anyelse can help me finish of the questions for 4,6,7,8 the same quality as earboth.

    Thank you and your help will be appreciated.
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  7. #7
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by novadragon849 View Post
    wow earboth your amazing. Anyway I was wondering if anyelse can help me finish of the questions for 4,6,7,8 the same quality as earboth.

    Thank you and your help will be appreciated.
    it would be hard to post solutions of the same quality as earboth, at least for me. you see, long division of polynomials is no problem, but typing it is hell, i don't know what progam earboth used to do that.

    how about this? i can give you hints as to what to do (the processes for the remaining ones are very similar to what earboth already did), you do them, and tell us your solutions. if they are wrong, someone will tell you why and how to fix it. it will also give you a chance to assess just how much you understood what earboth did
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  8. #8
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    well I will like to do 4,6,7,8 but I don't know how, so hows about you do a step by step for these questions and then I will do a past paper to see if I understood and I will post here if any issues arises.

    Sorry I meant to have the same step by step feature like earboth, not actually having the program.

    Thank you
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  9. #9
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    Quote Originally Posted by Jhevon View Post
    it would be hard to post solutions of the same quality as earboth, at least for me. you see, long division of polynomials is no problem, but typing it is hell, i don't know what progam earboth used to do that....

    Hello, jhevon,

    I don't use a special program even though it looks like that.

    You can use the [code ] [/code ] tag (but without the blank space in the brackets!) to write a text with equidistance between numbers or signs. So you only have to count the spaces to type terms in exact columns.

    EB
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  10. #10
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    can anyone help me?
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  11. #11
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    Quote Originally Posted by novadragon849 View Post
    ...
    8. Find the remainder when 2X^2 + 4X - 5 is divided by (2X+1)
    ...

    Hi,

    do long division:
    Code:
    (2x^2 + 4x - 5)  (2x +1) = x+3/2-(13/2)/(2x+1)
     2x^2 +  x
    ----------
           3x - 5
           3x + 3/2
          ---------
               -13/2
    So the remainder is -13/2

    EB
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