document.write( "Question 697017: For how many positive integers 1≤k≤1000 is the polynomial f_k(x)=x^3+x+k irreducible?
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Algebra.Com's Answer #429633 by Edwin McCravy(20055)\"\" \"About 
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document.write( "We find all the values of k when x³+x+k is reducible\r\n" );
document.write( "and subtract from 1000.\r\n" );
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document.write( "If x³+x+k is reducible over the polynomials\r\n" );
document.write( "with integer coefficients, then it is factorable as\r\n" );
document.write( "the product of a linear binomial and a quadratic\r\n" );
document.write( "trinomial both with leading coefficients 1.  That\r\n" );
document.write( "is, integers A,B,C exist such that:\r\n" );
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document.write( "x³+x+k = (x+A)(x²+Bx+C)\r\n" );
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document.write( "Multiplying the right side out we get\r\n" );
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document.write( "         (x+A)(x²+Bx+C) = x³+(A+B)x²+(AB+C)x+AC\r\n" );
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document.write( "So we have the identity:\r\n" );
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document.write( "x³+(A+B)x²+(AB+C)x+AC ≡ x³+x+k \r\n" );
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document.write( "So we can equate coefficients:\r\n" );
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document.write( "(1)     A+B= 0\r\n" );
document.write( "(2)   AB+C = 1\r\n" );
document.write( "(3)     AC = k\r\n" );
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document.write( "So from (1), B = -A, and substituting for B in (2),\r\n" );
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document.write( "A(-A)+C = 1\r\n" );
document.write( "  -A²+C = 1\r\n" );
document.write( "(4)   C = A²+1\r\n" );
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document.write( "So the factorization\r\n" );
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document.write( "x³+x+k = (x+A)(x²+Bx+C)\r\n" );
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document.write( "becomes\r\n" );
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document.write( "x³+x+k = (x+A)(x²-Ax+A²+1)\r\n" );
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document.write( "Substituting for C from (4), in (3),\r\n" );
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document.write( "A(A²+1) = k\r\n" );
document.write( "   A³+A = k\r\n" );
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document.write( "Since  1 ≤ k ≤ 1000\r\n" );
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document.write( "1 ≤ A³+A ≤ 1000\r\n" );
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document.write( "A³+A is a strictly increasing function, therefore:\r\n" );
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document.write( "The minimum value of A is 1, when k = A³+A = 1³+1 = 1+1 = 2, and\r\n" );
document.write( "The maximum value of A is 9, when k = A³+A = 9³+9 = 729+9 = 738.\r\n" );
document.write( "(For when A = 10, k = A³+A = 10³+10 = 1000+10 = 1010, which is\r\n" );
document.write( "over 1000.)\r\n" );
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document.write( "So there are 9 values of A, 1 through 9, and therefore 9  \r\n" );
document.write( "factorizations which are:\r\n" );
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document.write( "For k = 1, (x³+x+2) =(x+1)(x²-x+2)\r\n" );
document.write( "For k = 2, (x³+x+10) =(x+2)(x²-2x+5)\r\n" );
document.write( "For k = 3, (x³+x+30) =(x+3)(x²-3x+10)\r\n" );
document.write( "For k = 4, (x³+x+68) =(x+4)(x²-4x+17)\r\n" );
document.write( "For k = 5, (x³+x+130) =(x+5)(x²-5x+26)\r\n" );
document.write( "For k = 6, (x³+x+222) =(x+6)(x²-6x+37)\r\n" );
document.write( "For k = 7, (x³+x+350) =(x+7)(x²-7x+50)\r\n" );
document.write( "For k = 8, (x³+x+520) =(x+8)(x²-8x+65)\r\n" );
document.write( "For k = 9, (x³+x+738) =(x+9)(x²-9x+82)\r\n" );
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document.write( "Since for only 9 positive integers 1 ≤ k ≤ 1000, the \r\n" );
document.write( "polynomial f_k(x)=x³+x+k is reducible, then for the other\r\n" );
document.write( "1000-9 or 991 positive integers 1 ≤ k ≤ 1000, the \r\n" );
document.write( "polynomial f_k(x)=x³+x+k is irreducible.\r\n" );
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document.write( "Answer: 991\r\n" );
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document.write( "Edwin
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