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- Feb 14, 2012

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Factor $x^8+4x^2+4$ into two non-constant polynomials with integer coefficients.

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- Thread starter
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- #1

- Feb 14, 2012

- 3,894

-----

Factor $x^8+4x^2+4$ into two non-constant polynomials with integer coefficients.

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!

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- Feb 14, 2012

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1. Olinguito

2. Opalg

3. kaliprasad

Solution from Olinguito :

$=\ (x^8+4x^6+6x^4+4x^2+1)-4x^6-6x^4+3$

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

$=\ [(x^2+1)^4+2(x^2+1)^2+1]-4x^6-8x^4-4x^2$

$=\ [(x^2+1)^2+1]^2-4x^2(x^4+2x^2+1)$

$=\ [(x^2+1)^2+1]^2-4x^2(x^2+1)^2$

$=\ ([(x^2+1)^2+1]-2x(x^2+1))([(x^2+1)^2+1]+2x(x^2+1))$

$=\ (x^4-2x^3+2x^2-2x+2)(x^4+2x^3+2x^2+2x+2)$.

Alternate solution from Opalg :

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