document.write( "Question 1156301: Let P(x)=x4 −2x3 −10x2 +6x+45
\n" ); document.write( "▪ Use the Rational Zero Theorem to list all the possible rational zeros. ▪ Then find all zeros exactly (rational, irrational, and imaginary).
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Algebra.Com's Answer #778996 by ikleyn(52787) $\"\"$ $\"About$

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document.write( "Since the leading coefficient is 1, the Remainder theorem provides this list of possible zeros \r\n" );
document.write( "(all of them are divisors of the constant term 45, in this case)\r\n" );
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document.write( "   +/-1, +/-3, +/-5, +/-9, +/-15, +/-45.\r\n" );
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document.write( "Next, the plot below\r\n" );
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document.write( "    \r\n" );
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document.write( "    Plot y = \r\n" );
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document.write( "shows the root x= 3 of the multiplicity at least 2.\r\n" );
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document.write( "So, I divide    by  ,  and I get the quotient  .\r\n" );
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document.write( "This quotient is a quadratic polynomial with negative discriminant, so it has no real roots.\r\n" );
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document.write( "Therefore, factoring over real numbers is\r\n" );
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document.write( "     = .\r\n" );
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document.write( "The quadratic polynomial  x^2 + 4x + 5  has no rational roots.\r\n" );
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document.write( "It has no real roots, too, since its discriminant d = (-4)^2 - 4*1*5 = 16 - 20 = -4 is negative.\r\n" );
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document.write( "It has two complex roots   = .\r\n" );
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document.write( "ANSWER.  The roots of the given polynomial are  x= 3  of the multiplicity  2  and \r\n" );
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document.write( "         two complex roots    and     of the multiplicity 1 each.\r\n" );
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