document.write( "Question 1156241: Let P(x)=x4 −2x3 −10x2 +6x+45
\n" ); document.write( "▪ Use the Rational Zero Theorem to list all the possible rational zeros.
\n" ); document.write( "▪ Then find all zeros exactly (rational, irrational, and imaginary).
\n" ); document.write( "Hint: Use the Rational Zero Theorem, a graphing calculator, and synthetic division if needed. \n" ); document.write( "

Algebra.Com's Answer #778935 by ikleyn(52781) $\"\"$ $\"About$

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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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\n" ); document.write( "\n" ); document.write( "shows the root x= 3 of the multiplicity at least 2.\r
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\n" ); document.write( "\n" ); document.write( "So, I divide $\"x%5E4+-+2x%5E3+-+10x%5E2+%2B+6x+%2B+45\"$ by $\"%28x-3%29%5E2\"$ , and I get the quotient $\"x%5E2+%2B+4x+%2B+5\"$ .\r
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\n" ); document.write( "\n" ); document.write( "This quotient is a quadratic polynomial with negative discriminant, so it has no real roots.\r
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\n" ); document.write( "\n" ); document.write( "Therefore, factoring over real numbers is\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E4+-+2x%5E3+-+10x%5E2+%2B+6x+%2B+45\"$ = $\"%28x-3%29%5E2%2A%28x%5E2+%2B+4x+%2B+5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "You may go forward to find complex zeroes of the quadratic quotient.
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