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Math Help - Factoring problem

  1. #1
    Newbie
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    Feb 2008
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    Factoring problem

    Hi, guys! I really need the solution for this problem so please help!
    Factor the expression:
    a^4+2a^3b-3a^2b^2-4ab^3+4b^4
    The answer is (a-b)^2(a+2b)^2(according to my book).
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  2. #2
    Super Member

    Joined
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    Lexington, MA (USA)
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    Hello, brainmixer!

    Here's one of the ways to factor this polynomial . . .


    Factor: . a^4+2a^3b-3a^2b^2-4ab^3+4b^4

    Factor by grouping:

    . . a^2(a^2+2ab - 3b^2) - 4b^3(a - b)

    . . = \;a^2{\color{blue}(a-b)}(a+3b) - 4b^3{\color{blue}(a-b)}

    . . = \;(a-b)\cdot\bigg[a^2(a+3b) - 4b^3\bigg]

    . . = \;(a-b)\cdot\bigg[a^3 {\color{red}+ \,3a^2b} - 4b^3\bigg]

    . . = \;(a-b) \cdot\bigg[a^3 {\color{red}-\,a^2b + 4a^2b} - 4b^3\bigg]

    . . = \;(a-b)\cdot\bigg[a^2(a-b) + 4b(a^2-b^2)\bigg]

    . . = \;(a-b)\cdot\bigg[a^2{\color{blue}(a-b)} + 4b{\color{blue}(a-b)}(a+b)\bigg]

    . . = \;(a-b)\cdot(a-b)\cdot\bigg[a^2 + 4b(a+b)\bigg]

    . . = \;(a-b)^2\cdot[a^2+4ab+b^2]

    . . = \;(a-b)^2\cdot(a+2b)^2

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