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Math Help - Long Factoring

  1. #1
    rcs
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    Long Factoring

    x^4 + 2x^3 + 3x^2 + 2x + 1
    x ^4 + x^2 + 2x^3 + 2x^2 + 2x + 1
    (x^4 + 2x^3 + x^2) +( 2x^2 + 2x +1)
    x^2 ( x^2 + 2x + 1) + 2 (x^2 + x) + 1
    x^2(x+1)^2 + 2x (x+1) + 1
    [x(x+1)+ 1]^2 + 2[x(x+1)] + 1
    [x(x+1) + 1]^2
    (x^2 + x + 1)^2 answer.
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  2. #2
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    Hello, rcs!

    Your work is absolutely correct! . . . Good work!



    \text{Factor: }\:x^4 + 2x^3 + 3x^2 + 2x + 1

    The coefficients reminded me of an arithmetical phenomenon:

    . . . \begin{array}{ccc} 1^2 &=& 1 \\ 11^2 &=& 121 \\ 111^2 &=& 12321 \\ 1111^2 &=& 1234321 \\ \vdots && \vdots \end{array}

    So I immediately concluded that we have: . (x^2 + x + 1)^2


    Already knowing the answer, it was easy to demonstrate the factoring:

    x^4 + 2x^3 + 3x^2 + 2x + 1

    . . =\;(x^4 + x^3 + x^2) + (x^3 + x^2 + x) + (x^2 + x + 1)

    . . =\;x^2(x^2+x+1) + x(x^2+x+1) + (x^2+x+1)

    . . =\;(x^2+x+1)(x^2+x+1)

    . . =\;(x^2+x+1)^2

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  3. #3
    rcs
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    thank you

    thank you so much Soroban. i was trying to solve ... thank God you checked my work. Love Live MHF!
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