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anonymous

  • one year ago

Bet you cant solve this.

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  1. anonymous
    • one year ago
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    For how many integers n with |n| <500 can the polynomial \[Pn(x)=x^{6}+n\] be written as a product of two non-constant polynomials with integer coefficients?

  2. anonymous
    • one year ago
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    your right

  3. anonymous
    • one year ago
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    STOP CHEATING LOL @JoannaBlackwelder

  4. anonymous
    • one year ago
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    its 0 ofc.

  5. anonymous
    • one year ago
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    would attack this problem along these lines which you can refine. If the polynomial Pn(x) = (x6 + n) could be written as a product of 2 non-constant polynomials with integer coefficients, then by the Fundamental Theorem of Algebra one of those factors would represent a real root when Pn(x) = 0. Let's plot Pn(x). First x6 is basically a parabola going thru (x,y) = (0,0) that opens upward and is symmetric about the y-axis. The + n is a positive or negative integer that raises or lowers the parabola along the y-axis leaving x alone. Hence, for n>=+1, Pn(x) cannot have roots since it always has positive y, and so cannot be expresses in the product format. For n=0, you get the degenerate case of x6 = 0 with 6 identical x=0 roots. so you can represent x6 = x1x5, x2x4 etc. For n negative integers, the parabola gets lowered so there are exactly 2 distinct real integer roots: If n = -1, the roots are x = +- 1 and a factorization is possible If n = -64, the roots are x = +- 2 and a factorization is possible If x = +-3, then n would be -729 = - (3)6, which exceeds your |n| < 500 constraint.

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