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Math Help - Factorising this polynomial over C

  1. #1
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    Factorising this polynomial over C

    Hi there

    I'm having trouble getting this problem out.
    I have to factorise this over c.
    4z^2 - 4z + 17
    My answer is:
    z^2 - z + \frac{17}{4}
    (z^2 - z + \frac{1}{4}) +  \frac{17}{4} - \frac{1}{4}
    (z - \frac{1}{2})^2 +  \frac{16}{4}
    (z - \frac{1}{2})^2 +  4
    (z - \frac{1}{2})^2 -  (2i)^2
    (z - \frac{1}{2} -  2i)(z - \frac{1}{2} +  2i)

    However, the answer in the book is this:
    (2z - 1 +  4i)(2z - 1 -  4i)
    Is this the same as what I had?
    Also, have I put this complex number question in the right forum? What section should a question like this go?
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  2. #2
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    Quote Originally Posted by Bucephalus View Post
    Hi there

    I'm having trouble getting this problem out.
    I have to factorise this over c.
    4z^2 - 4z + 17
    My answer is:
    z^2 - z + \frac{17}{4}
    (z^2 - z + \frac{1}{4}) +  \frac{17}{4} - \frac{1}{4}
    (z - \frac{1}{2})^2 +  \frac{16}{4}
    (z - \frac{1}{2})^2 +  4
    (z - \frac{1}{2})^2 -  (2i)^2
    (z - \frac{1}{2} -  2i)(z - \frac{1}{2} +  2i)

    However, the answer in the book is this:
    (2z - 1 +  4i)(2z - 1 -  4i)
    Is this the same as what I had?
    Also, have I put this complex number question in the right forum? What section should a question like this go?
    You lost a 4 in your very first step

    \displaystyle 4\left(z^2 - z + \frac{17}{4} \right)

    Now in your final answer if you break 4=2\cdot 2

    and put it back into each factor your answers will be the same.
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  3. #3
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    Quote Originally Posted by Bucephalus View Post
    Hi there

    I'm having trouble getting this problem out.
    I have to factorise this over c.
    4z^2 - 4z + 17
    My answer is:
    z^2 - z + \frac{17}{4}
    (z^2 - z + \frac{1}{4}) + \frac{17}{4} - \frac{1}{4}
    (z - \frac{1}{2})^2 + \frac{16}{4}
    (z - \frac{1}{2})^2 + 4
    (z - \frac{1}{2})^2 - (2i)^2
    (z - \frac{1}{2} - 2i)(z - \frac{1}{2} + 2i)

    However, the answer in the book is this:
    (2z - 1 + 4i)(2z - 1 - 4i)
    Is this the same as what I had?
    Also, have I put this complex number question in the right forum? What section should a question like this go?
    Your coefficient of z^2 is 1. The coefficient in the question is 4.

    I suggest the following:

    4z^2 - 4z + 17 = 4(z^2 -z) + 17 = 4\left(\left[z - \frac{1}{2}\right]^2 - \frac{1}{4}\right) + 17 = (2z - 1)^2 + 16 = ....
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