document.write( "Question 1188992: To divide the function f(x) by (x^2-4x-12), we find that the quotient is Q(x) and the remainder is (x+6). If f(-2)=a and f(6)=b, what is the value of b/a? \n" ); document.write( "

Algebra.Com's Answer #820196 by ikleyn(52788) $\"\"$ $\"About$

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\n" ); document.write( "To divide the function f(x) by (x^2-4x-12), we find that the quotient is Q(x)
\n" ); document.write( "and the remainder is (x+6). If f(-2)=a and f(6)=b, what is the value of b/a?
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\n" ); document.write( "\n" ); document.write( " First of all, the problem is worded incorrectly. To be correct, it must sound THIS WAY:\r
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document.write( "            To divide the  POLYNOMIAL f(x) by (x^2-4x-12), we find that the quotient is Q(x) \r\n" );
document.write( "            and the remainder is (x+6). If f(-2)=a and f(6)=b, what is the value of b/a?\r\n" );
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\n" ); document.write( "\n" ); document.write( " With this correction, see my solution below.\r
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document.write( "From the condition, we have this polynomial equality\r\n" );
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document.write( "    f(x) = (x^2-4x-12)*Q(x) + (x+6),    (1)\r\n" );
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document.write( "where Q(x) is some polynomial (the quotient of the division).\r\n" );
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document.write( "    Notice that the polynomial  (x^2 -4x - 12)  has the roots -2 and 6: \r\n" );
document.write( "        exactly the values mentioned in the problem's condition.\r\n" );
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document.write( "THEREFORE, substituting the values  x= -2  and  x= 6 into formula (1), we get\r\n" );
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document.write( "     a = f(-2) = 0*Q(-2) + (-2+6) = 0 + 4  = 4,\r\n" );
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document.write( "     b = f(6)  = 0*Q(6)  + (6+6)  = 0 + 12 = 12.\r\n" );
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document.write( "Hence,   =  = 3.\r\n" );
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document.write( "ANSWER.  The value of    is 3.\r\n" );
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