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Whats the best way to factor this

  1. Sep 6, 2014 #1
    1. The problem statement, all variables and given/known data

    2cos^2(θ) + cos(θ) - 1

    2. Relevant equations

    2cos^2(θ) + cos(θ) - 1

    3. The attempt at a solution

    the answer is (2cos(θ)-1)(cos(θ)+1)

    How do I factor this to give me that answer?
     
  2. jcsd
  3. Sep 6, 2014 #2

    Student100

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    Education Advisor
    Gold Member

    Do a substitution, x = cos(theta)

    Now you have 2x^2 + x -1.

    You know the signs of the factors must be a plus and a minus, so ( + ) ( - ) the two on your x squared lets you know that you need a two x. And the 1 tells you both end digits must be 1. (2x-1) (x +1) is the only combination that works to leave you with one quantity of positive x.
     
  4. Sep 7, 2014 #3
    Here's an outline of the solution: As pointed out by Student100 the substitution ##\cos\theta\leadsto x## produces a quadratic, and if you know the roots of a quadratic -- say ##ax^2+bx+c## with roots ##r_1,r_2## if any -- then you can factor it as ##a(x-r_1)(x-r_2)##. The only step that remains is to “undo” the substitution by replacing ##x## with the original ##\cos\theta##. I hope you can achieve all of those steps to find your desired answer. ;-)
     
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