document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=251543>Question 251543</A>: <I><SPAN STYLE=\"word-break: break-all;\"> how would you find \"p\" in the polynomial function:<BR>\n" );
document.write( "8x^3+4x^2-\"p\"x+6  ...is divided by 2x-1  ....the remainder is 3  ....determine the value of p!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #183222 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=183222\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First write the problem out in long division form\r<BR>\n" );
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document.write( "    2x-1 | 8x^3+4x^2-px+6\r\n" );
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document.write( "Now ask yourself: what must you multiply 2x by to get 8x^3? The answer is 4x^2. Place this above the x^2 term\r<BR>\n" );
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document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "         _______4x^2__________\r\n" );
document.write( "    2x-1 | 8x^3+4x^2-px+6\r\n" );
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document.write( "Multiply 4x^2 by 2x-1 to get 8x^2-4x^2 and place that beneath the first two terms\r<BR>\n" );
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document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "         _______4x^2__________\r\n" );
document.write( "    2x-1 | 8x^3+4x^2-px+6\r\n" );
document.write( "           8x^3-4x^2\r\n" );
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document.write( "Subtract the bottom expression 8x^3-4x^2 from the top expression 8x^3+4x^2 to get 8x^2 and bring down the rest of the terms.\r<BR>\n" );
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document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "         _______4x^2__________\r\n" );
document.write( "    2x-1 | 8x^3+4x^2-px+6\r\n" );
document.write( "           8x^3-4x^2\r\n" );
document.write( "           ---------\r\n" );
document.write( "                8x^2-px+6\r\n" );
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document.write( "This is where we start back from the beginning and ask: what expression can you multiply 2x by to get 8x^2? The answer is 4x. Place this above the -px term\r<BR>\n" );
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document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "         _______4x^2+4x__________\r\n" );
document.write( "    2x-1 | 8x^3+4x^2-px+6\r\n" );
document.write( "           8x^3-4x^2\r\n" );
document.write( "           ---------\r\n" );
document.write( "                8x^2-px+6\r\n" );
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document.write( "Multiply 4x by 2x-1 to get 8x^2-4x. Place this underneath 8x^2-px+6\r<BR>\n" );
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document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "         _______4x^2+4x__________\r\n" );
document.write( "    2x-1 | 8x^3+4x^2-px+6\r\n" );
document.write( "           8x^3-4x^2\r\n" );
document.write( "           ---------\r\n" );
document.write( "                8x^2-px+6\r\n" );
document.write( "                8x^2-4x\r\n" );
document.write( "                ---------\r\n" );
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document.write( "Subtract 8x^2-4x from 8x^2-px+6\r<BR>\n" );
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document.write( "         _______4x^2+4x__________\r\n" );
document.write( "    2x-1 | 8x^3+4x^2-px+6\r\n" );
document.write( "           8x^3-4x^2\r\n" );
document.write( "           ---------\r\n" );
document.write( "                8x^2-px+6\r\n" );
document.write( "                8x^2-4x\r\n" );
document.write( "                ---------\r\n" );
document.write( "                    -px+4x+6\r\n" );
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document.write( "Combine like terms.\r<BR>\n" );
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document.write( "         _______4x^2+4x__________\r\n" );
document.write( "    2x-1 | 8x^3+4x^2-px+6\r\n" );
document.write( "           8x^3-4x^2\r\n" );
document.write( "           ---------\r\n" );
document.write( "                8x^2-px+6\r\n" );
document.write( "                8x^2-4x\r\n" );
document.write( "                ---------\r\n" );
document.write( "                    (-p+4)x+6\r\n" );
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document.write( "Now onto the final stage: what expression must you multiply 2x by to get (-p+4)x? The answer is (1/2)(-p+4). Write this above the last term 6\r<BR>\n" );
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document.write( "         _______4x^2+4x+(1/2)(-p+4)__________\r\n" );
document.write( "    2x-1 | 8x^3+4x^2-px+6\r\n" );
document.write( "           8x^3-4x^2\r\n" );
document.write( "           ---------\r\n" );
document.write( "                8x^2-px+6\r\n" );
document.write( "                8x^2-4x\r\n" );
document.write( "                ---------\r\n" );
document.write( "                    (-p+4)x+6\r\n" );
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document.write( "Multiply (1/2)(-p+4) by 2x-1 to get (-p+4)x-(1/2)(-p+4)\r<BR>\n" );
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document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "         _______4x^2+4x+(1/2)(-p+4)__________\r\n" );
document.write( "    2x-1 | 8x^3+4x^2-px+6\r\n" );
document.write( "           8x^3-4x^2\r\n" );
document.write( "           ---------\r\n" );
document.write( "                8x^2-px+6\r\n" );
document.write( "                8x^2-4x\r\n" );
document.write( "                ---------\r\n" );
document.write( "                    (-p+4)x+6\r\n" );
document.write( "                    (-p+4)x-(1/2)(-p+4)\r\n" );
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document.write( "Subtract the terms and distribute.\r<BR>\n" );
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document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "         _______4x^2+4x+(1/2)(-p+4)__________\r\n" );
document.write( "    2x-1 | 8x^3+4x^2-px+6\r\n" );
document.write( "           8x^3-4x^2\r\n" );
document.write( "           ---------\r\n" );
document.write( "                8x^2-px+6\r\n" );
document.write( "                8x^2-4x\r\n" );
document.write( "                ---------\r\n" );
document.write( "                    (-p+4)x+6\r\n" );
document.write( "                    (-p+4)x-(1/2)(-p+4)\r\n" );
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document.write( "                          6-(1/2)p+2\r\n" );
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document.write( "Combine like terms.\r<BR>\n" );
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document.write( "         _______4x^2+4x+(1/2)(-p+4)__________\r\n" );
document.write( "    2x-1 | 8x^3+4x^2-px+6\r\n" );
document.write( "           8x^3-4x^2\r\n" );
document.write( "           ---------\r\n" );
document.write( "                8x^2-px+6\r\n" );
document.write( "                8x^2-4x\r\n" );
document.write( "                ---------\r\n" );
document.write( "                    (-p+4)x+6\r\n" );
document.write( "                    (-p+4)x-(-p+4)\r\n" );
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document.write( "                          8-(1/2)p\r\n" );
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document.write( "So the remainder is 8-(1/2)p. But we're given that the remainder is 3. So this means that 8-(1/2)p=3 and its solution is p=10.\r<BR>\n" );
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document.write( "So the value of p is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p=10 ALIGN=MIDDLE  ALT=\"p=10\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So dividing the polynomial <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x%5E3%2B4x%5E2-10x%2B6 ALIGN=MIDDLE  ALT=\"8x%5E3%2B4x%5E2-10x%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" > by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-1 ALIGN=MIDDLE  ALT=\"2x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > will give you some quotient with a remainder of 3.\r<BR>\n" );
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