Algebra.Com's Answer #595554 by josgarithmetic(39620)![\"\"](\"/images/stars/stars-13.gif\")  
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Factoring your first polynomial and your H(x) function are beyond the typical Intermediate
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document.write( "Algebra level. Make use of Rational Roots Theorem.\r
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document.write( " can be tested for possible roots, plus and minus 1,2,3,6. This can be done with
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document.write( "synthetic division. You could expect at most, three roots.
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document.write( " is a root, giving remainder 0, and coefficients of quotient, 1,2,-3.
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document.write( " is a root, giving remainder 0, and coefficients of quotient, 1,-1.
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document.write( "This gives you a factorization 
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document.write( "The meaning of those last coefficients of 1,-1 is  , or as shown,  .\r
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document.write( " may have any roots (need to test each) of again, plus and minus 1,2,3,6.
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document.write( "Testing for roots plus and minus 1,2,3 seem to give nonzero remainders. Maybe continue testing
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document.write( "for -6 and +6....
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document.write( "Neither give 0 remainder, so no rational roots.
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document.write( "H(x) may have irrational roots, but checking for them is harder. Other tests are possible
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document.write( "but are more lengthy.
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document.write( "Using a graphing tool, like putting y=10x^4+2x^2-x+6 into the text field of Google search engine
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document.write( "shows a fairly symmetric looking U shaped graph, and the minimum is above the x-axis everywhere,
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document.write( "so H(x) has no real roots. This also means no real irrational roots.
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document.write( "Note that there ARE more roots to check. numerators are plus or minus 1,2,3,6 and denominators
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document.write( "are plus and minus 1,2,5,10. I did not try the possible roots other than those for whole numbers.
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document.write( "You should expect up to FOUR complex roots.\r
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document.write( "You can try Rational Roots Theorem the same way for  .
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document.write( "Roots to test for, using synthetic division, are plus and minus 1/2, 1/3, 1/4, 1/6, maybe 1/12.
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document.write( "The negatives of those are not roots, but still you should expect up to five complex roots, since h(x) is degree 5.
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document.write( "(More work is necessary to find them).
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document.write( "(Graphing tool indicates ONE negative real root approximately  ).
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