document.write( "Question 1017245: How many roots do the following equations have? -12x^2 -25x+5+x^3=0 \n" ); document.write( "

Algebra.Com's Answer #633569 by ikleyn(52787) $\"\"$ $\"About$

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\n" ); document.write( "How many roots do the following equations have? -12x^2 -25x+5+x^3=0
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document.write( "1. In the complex plane, this polynomial has three roots, exactly as its degree is.\r\n" );
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document.write( "2. In the real domain, it definitely has at least one root.\r\n" );
document.write( "   You may conclude it, because the degree of the polynomial is 3 and it tends to  as x --->  and tends to  as x ---> .\r\n" );
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document.write( "3. Now let us check the sign of the polynomial at selected points:\r\n" );
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document.write( "   f(-1) = -12*(-1)^2 - 25*(-1) + 5 - (-1)^3 = 12 + 25 + 5 + 1 = 43 > 0.\r\n" );
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document.write( "   f(1) = -12*1^2 - 25*1 + 5 - 1^3 = -12 - 25 + 5 - 1 = -33 <0.\r\n" );
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document.write( "   It is just enough to conclude that f(x) has three real roots:\r\n" );
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document.write( "      one between  and (-1),     <--- the polynomial change the sign at the ends of this interval;\r\n" );
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document.write( "      second between (-1) and 1,      <---  again, the polynomial change the sign at the ends of this interval;\r\n" );
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document.write( "      third between 1 and .        <--- by the same reason.\r\n" );
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