document.write( "Question 530888: Use the intermediate value theorem to show that f(x) has a zero in the given interval. Please show all of your work.
\n" ); document.write( "f(x)= -3x^4-5x^3-8x+7; [-2.4,-2.3] \n" ); document.write( "
Algebra.Com's Answer #350254 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Since f(x) is a polynomial, we only have to show that \r\n" );
document.write( "f(-2.4) and f(-2.3) have different signs. \r\n" );
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document.write( "There are two ways to find f(-2.4) and f(-2.3)\r\n" );
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document.write( "Method I.  Direct substitution:\r\n" );
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document.write( "f(-2.4) = -3(-2.4)* - 5(-2.4)³ - 8x + 7 \r\n" );
document.write( "        = -3(33.1776) - 5(-13.824) - 8(-2.4) + 7\r\n" );
document.write( "        = -99.5328 + 69.12 + 19.2 + 7\r\n" );
document.write( "        = -4.2128 \r\n" );
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document.write( "f(-2.3) = -3(-2.3)* - 5(-2.3)³ - 8x + 7 \r\n" );
document.write( "        = -3(27.9841) - 5(-12.9841) - 8(-2.3) + 7\r\n" );
document.write( "        = -83.9523 + 60.835 + 18.4 + 7\r\n" );
document.write( "        =  2.2827\r\n" );
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document.write( "They have different signs, so that's all you need to guarantee\r\n" );
document.write( "that a polynomial has a zero in the interval between the two\r\n" );
document.write( "values of x. \r\n" );
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document.write( "Method II.  The remainder theorem (synthetic division)\r\n" );
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document.write( "-2.4 | -3  -5        0  -8        7\r\n" );
document.write( "     |      7.2  -5.28  12.672  -11.2128\r\n" );
document.write( "       -3   2.2  -5.28   4.672   -4.2128\r\n" );
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document.write( "So f(-2.4) = the remainder = -4.2128\r\n" );
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document.write( "-2.3 | -3  -5        0  -8       7\r\n" );
document.write( "     |      6.9  -4.37  10.051  -4.7173\r\n" );
document.write( "       -3   1.9  -4.37   2.051   2.2827\r\n" );
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document.write( "So f(-2.3) = the remainder = 2.2827\r\n" );
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document.write( "They have different signs, so that's all you need to guarantee\r\n" );
document.write( "that a polynomial has a zero in the interval between the two\r\n" );
document.write( "values of x.\r\n" );
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document.write( "Edwin
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