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Math Help - Factorising

  1. #1
    Senior Member
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    Factorising

    How would I factor these:

    1) x^7+x

    2) x^12 - y^12

    And I'm also having trouble with this problem:

    1) If x+y=1 and x^3+y^3=19. What is x^2+y^2
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  2. #2
    Senior Member topher0805's Avatar
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    1) divide by x to get: x(1+6x)

    2) this is like a really complicated difference of squares. x^2-y^2 = (x+y)(x-y), etc.

    3) use the first equation to get x = 1 - y, then plug that into the second equation to figure out y. Once you have y, plug it into the first equation to find x, then plug them both into x^2 + y^2 to get your answer.
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  3. #3
    Senior Member DivideBy0's Avatar
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    Quote Originally Posted by acevipa View Post
    How would I factor these:

    1) x^7+x

    2) x^12 - y^12

    And I'm also having trouble with this problem:

    1) If x+y=1 and x^3+y^3=19. What is x^2+y^2
    Question 1

    x^7+x=x(x^6+1)=x((x^2)^3+(1)^3)=x(x^2+1)(x^4-x^2+1)


    Question 2

    x^{12}-y^{12}=(x^6-y^6)(x^6+y^6)=((x^2)^3+(y^2)^3)(x^3-y^3)(x^3+y^3)

    =(x^2+y^2)(x^4-x^2y^2+y^4)(x-y)(x^2+xy+y^2)(x+y)(x^2-xy+y^2)

    Last Question

    (x+y)^3=x^3+y^3+3xy(x+y)

    Subbing in values:

    1^3=19+3xy(1) \Leftrightarrow 3xy=-18 \Leftrightarrow xy=-6

    Also, (x+y)^2=x^2+y^2+2xy

    Subbing in again,

    (1)^2=x^2+y^2+2(-6)

    \Rightarrow x^2+y^2=13
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