document.write( "Question 1042747: Use the Theorem for bounds on zeros to find a bound on the real zeros of the polynomial function.\r
\n" ); document.write( "\n" ); document.write( "f(x) = x^4 + 2x^2 - 3
\n" ); document.write( " a. -4 and 4
\n" ); document.write( " b. -3 and 3
\n" ); document.write( " c. -6 and 6
\n" ); document.write( " d. -5 and 5 \n" ); document.write( "

Algebra.Com's Answer #657796 by Edwin McCravy(20055) $\"\"$ $\"About$

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f(x) = x⁴ + 2x² - 3
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document.write( "Upper Bound: \r\n" );
document.write( "If we divide a polynomial function f(x) by (x-r), where r is\r\n" );
document.write( "POSITIVE using synthetic division, and we get all positive \r\n" );
document.write( "numbers on the bottom row of the synthetic division, then r is \r\n" );
document.write( "an upper bound to the real zeros of f(x).\r\n" );
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document.write( "Lower Bound: \r\n" );
document.write( "If we divide a polynomial function f(x) by (x-r), where r is\r\n" );
document.write( "NEGATIVE, using synthetic division, and we get alternating \r\n" );
document.write( "signs for the numbers on the bottom row of the synthetic\r\n" );
document.write( "division, then r is a lower bound to the real zeros of f(x). \r\n" );
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document.write( "We try the smallest positive number among the choices,\r\n" );
document.write( "which is 3\r\n" );
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document.write( "We use 3 to see if it is an upper bound to the zeros\r\n" );
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document.write( "3 |  1 0 2  0 -3\r\n" );
document.write( "  |    3 6 24 72\r\n" );
document.write( "     1 3 8 24 69   \r\n" );
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document.write( "All the numbers on the bottom row are positive, so 3 is\r\n" );
document.write( "an upper bound for the zeros of f(x).\r\n" );
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document.write( "We try the negative number -3 that goes with that choice\r\n" );
document.write( "to see if it is a lower bound:\r\n" );
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document.write( "-3 |  1  0  2   0 -3\r\n" );
document.write( "   |    -3  9 -33 99\r\n" );
document.write( "      1 -3 11 -33 96 \r\n" );
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document.write( "The signs on the bottom row of the synthetic division\r\n" );
document.write( "alternate so -3 is a lower bound.\r\n" );
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document.write( "So all the zeros, if any, are between -3 and +3.\r\n" );
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document.write( "But, for that matter, since they are between -3 and +3,\r\n" );
document.write( "they are also between -4 and +4.\r\n" );
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document.write( "They are also between -5 and +5\r\n" );
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document.write( "They are also between -6 and 6.\r\n" );
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document.write( "So actually every choice listed are also upper and lower\r\n" );
document.write( "bounds.  Maybe they asked for the LEAST ones in absolute \r\n" );
document.write( "value that are on the list.  But if not, the author of\r\n" );
document.write( "the book it came from has botched the problem. \r\n" );
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document.write( "You might point out to your teacher that ALL of the \r\n" );
document.write( "choices given above are bounds for the zeros.\r\n" );
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document.write( "The only zeros are -1 and 1.  So any two numbers such that\r\n" );
document.write( "the smaller number is less than -1 and the larger number is\r\n" );
document.write( "greater than +1 are bounds for the zeroes.  This is really\r\n" );
document.write( "a botched problem.  All the choices are correct!\r\n" );
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document.write( "Edwin

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