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Math Help - Factor of the polynomial

  1. #1
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    Factor of the polynomial

    a) Show that 2x-1 is a factor of the polynomial 2x - 3x - 11x + 6
    Find the other factors of the polynomial

    b) Find, correct to three decimal places, the values of t satisfying

    2e^3t - 3e^2t - 11e^t + 6 = 0



    Could someone hellp me start this question because I've no idea at all what to do. If someone could start me off i might be able to figure it out.

    thanks
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  2. #2
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    Quote Originally Posted by djmccabie View Post
    a) Show that 2x-1 is a factor of the polynomial 2x - 3x - 11x + 6
    Find the other factors of the polynomial

    b) Find, correct to three decimal places, the values of t satisfying

    2e^3t - 3e^2t - 11e^t + 6 = 0
    a) if 2x-1 is a factor, then f\left(\frac{1}{2}\right) = 0 ... show this by synthetic division

    Code:
    1/2]      2    -3    -11    6
    
                   1     -1    -6
    ------------------------------
             2    -2     -12    0
    the depressed polynomial is

    2x^2 - 2x - 12 = 2(x^2 - x - 6) = 2(x - 3)(x + 2)

    the other two roots are x = 3 and x = -2

    b) note the similarity between this equation and the one in part (a)

    e^t = \frac{1}{2}

    e^t = 3

    e^t = -2

    two of the three above equations will yield a real solution.
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  3. #3
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    Quote Originally Posted by skeeter View Post
    Code:
    1/2]      2    -3    -11    6
    
                   1     -1    -6
    ------------------------------
             2    -2     -12    0
    the depressed polynomial is

    2x^2 - 2x - 12 = 2(x^2 - x - 6) = 2(x - 3)(x + 2)

    Hi thanks for your help. I don't really understand this part of your answer. I've never learned what a depressed polynomial is. how did you deduce the 1 -1 -6 part? thanks
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  4. #4
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    Hi i just found a tutorial for synthetic division online. I've never even heard of it. Thanks!
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  5. #5
    Super Member 11rdc11's Avatar
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    Quote Originally Posted by djmccabie View Post
    Hi thanks for your help. I don't really understand this part of your answer. I've never learned what a depressed polynomial is. how did you deduce the 1 -1 -6 part? thanks
    Try googling synthetic division and it should make sense. I would explain to you but just not sure how to.
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  6. #6
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    Hello, djmccabie!

    a) Show that 2x-1 is a factor of the polynomial 2x^3 - 3x^2 - 11x + 6

    Find the other factors of the polynomial
    What's stopping you from using long division?


    . . \begin{array}{cccccc} & & & x^2 & -x & -6 \\<br />
& & -- & -- & -- & -- \\ 2x-1 & ) & 2x^3 & -3x^2 & -11x & +6 \\ & & 2x^3 & -x^2 \\ & & -- & -- \\ & & & -2x^2 & -11x \\ & & & -2x^2 & +x \\ & & & -- & -- \\ & & & & -12x & +6 \\ & & & & -12x & +6 \\ & & & & -- & --\end{array}



    Therefore: . 2x^3-3x^2-11x+6 \:=\:(2x-1)(x^2-x-6) \:=\:\boxed{(2x-1)(x+2)(x-3)}

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