document.write( "Question 1011730: The following function is given\r
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\n" ); document.write( "\n" ); document.write( "A. List all rational zeros that are possible according to the Rational Zero Theorem.\r
\n" ); document.write( "\n" ); document.write( "All rational zeros: \r
\n" ); document.write( "\n" ); document.write( "B. Use synthetic division to test several possible rational zeros in order to identify one actual zero\r
\n" ); document.write( "\n" ); document.write( "One rational zero of the given function is: \r
\n" ); document.write( "\n" ); document.write( "C. Use the zero from part (b) to find all the zeros of the polynomial function.\r
\n" ); document.write( "\n" ); document.write( "The zeros of the function f(x)=x^3 -5x^2 -9x+45 are: \n" ); document.write( "

Algebra.Com's Answer #627590 by Edwin McCravy(20056) $\"\"$ $\"About$

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document.write( "All potential rational zeros of a polynomial with only integer\r\n" );
document.write( "coefficients are numbers of the form ±P/Q where P is a divisor \r\n" );
document.write( "of the absolute value of the constant term and Q is a divisor of the \r\n" );
document.write( "absolute value of the leading coefficient. \r\n" );
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document.write( "f(x) = x³-5x²-9x+45\r\n" );
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document.write( "The constant term is 45\r\n" );
document.write( "The leading term is x³\r\n" );
document.write( "The leading coefficient is 1.\r\n" );
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document.write( "In this case since the leading coefficient is 1, which has only\r\n" );
document.write( "the divisor 1, we only need to consider as potential rational zeros\r\n" );
document.write( "± the divisors of 45, which are\r\n" );
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document.write( "±1, ±3, ±5, ±9, ±15, and ±45.\r\n" );
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document.write( "We try 1\r\n" );
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document.write( "1 | 1 -5  -9  45\r\n" );
document.write( "  |    1  -4 -13\r\n" );
document.write( "    1 -4 -13  32\r\n" );
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document.write( "The remainder is 32, not 0, so 1 is not a zero of f(x)\r\n" );
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document.write( "We try -1\r\n" );
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document.write( "-1 | 1 -5  -9  45\r\n" );
document.write( "   |   -1   6   4\r\n" );
document.write( "     1 -6  -4  48\r\n" );
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document.write( "The remainder is 48, not 0, so -1 is not a zero of f(x)\r\n" );
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document.write( "We try 3\r\n" );
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document.write( " 3 | 1 -5  -9  45\r\n" );
document.write( "   |    3  -6 -45\r\n" );
document.write( "     1 -2 -15   0\r\n" );
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document.write( "The remainder is 0, so 3 is a zero of f(x).\r\n" );
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document.write( "Since the synthetic division was actually a division\r\n" );
document.write( "of f(x) by x-3, we can use the numbers on the bottom\r\n" );
document.write( "line of the synthetic division to factor f(x) as\r\n" );
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document.write( "f(x) = (x-3)(x²-2x-15)\r\n" );
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document.write( "We only need to find the zeros of x²-2x-15 to find\r\n" );
document.write( "the other zeros, rational or otherwise of f(x).\r\n" );
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document.write( "We are able to do further factoring:\r\n" );
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document.write( "f(x) = (x-3)(x+3)(x-5)\r\n" );
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document.write( "Therefore all zeros of f(x) are found by setting each of\r\n" );
document.write( "these expressions = 0 and solving:\r\n" );
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document.write( "    x-3 = 0;   x+3 = 0;   x-5 = 0\r\n" );
document.write( "      x = 3      x = -3     x = 5\r\n" );
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document.write( "Thus there are three zeros, 3,-3, and -5.\r\n" );
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document.write( "In we graph of f(x) we see that these three zeros are the \r\n" );
document.write( "x-intercepts:\r\n" );
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document.write( "Edwin

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