document.write( "Question 1200922: All the real zeros of the given polynomial are integers. Find the zeros. (Enter your answers as a comma-separated list. Enter all answers including repetitions.)\r
\n" ); document.write( "\n" ); document.write( "P(x) = x^3 − 3x − 2\r
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\n" ); document.write( "\n" ); document.write( "Write the polynomial in factored form.\r
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Algebra.Com's Answer #835154 by ikleyn(52873) $\"\"$ $\"About$

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\n" ); document.write( "All the real zeros of the given polynomial are integers. Find the zeros.
\n" ); document.write( "(Enter your answers as a comma-separated list. Enter all answers including repetitions.)
\n" ); document.write( "P(x) = x^3 − 3x − 2
\n" ); document.write( "x =
\n" ); document.write( "Write the polynomial in factored form.
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document.write( "It is an easy case of the Rational Zeroes theorem (since the leading coefficient \r\n" );
document.write( "of the polynomial is 1 (one)).\r\n" );
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document.write( "According to this theorem, the list of possible rational zeroes consists of these values\r\n" );
document.write( "{1, -1, 2, -2}.\r\n" );
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document.write( "It is easy to check that x= -1 is the root.\r\n" );
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document.write( "Then the given polynomial is divisible by (x+1), so we divide the given polynomial\r\n" );
document.write( "by (x+1) to reduce the degree\r\n" );
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document.write( "     = x^2 - x - 2.\r\n" );
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document.write( "Regarding the quadratic polynomial  x^2 - x - 2, we can factor it mentally\r\n" );
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document.write( "    x^2 - x - 2 = (x-2)*(x+1).\r\n" );
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document.write( "Therefore, the final decomposition of the given polynomial is \r\n" );
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document.write( "    x^3 − 3x − 2 = .\r\n" );
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document.write( "It has the roots x= 2 of multiplicity 1 and x= -1 of multiplicity 2.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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