document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=149769>Question 149769</A>: <I><SPAN STYLE=\"word-break: break-all;\"> don't know how to check my amswers for these....\r<BR>\n" );
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document.write( "simplify: (3x^3-x^2-4x+1)/(x-1)\r<BR>\n" );
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document.write( "factor: 18x+x^2-11x\r<BR>\n" );
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document.write( "factor: 6x^2+x-2\r<BR>\n" );
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document.write( "factor: 12x^3+31x^2+20x </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #109869 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=109869\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> # 1\r<BR>\n" );
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document.write( "Let's simplify this expression using synthetic division\r<BR>\n" );
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document.write( "Start with the given expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%5E3+-+x%5E2+-+4x+%2B+1%29%2F%28x-1%29 ALIGN=MIDDLE  ALT=\"%283x%5E3+-+x%5E2+-+4x+%2B+1%29%2F%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "First lets find our test zero:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-1=0 ALIGN=MIDDLE  ALT=\"x-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the denominator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-1 ALIGN=MIDDLE  ALT=\"x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > equal to zero\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Solve for x.\r<BR>\n" );
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document.write( "so our test zero is 1\r<BR>\n" );
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document.write( "Now set up the synthetic division table by placing the test zero in the upper left corner and placing the coefficients of the numerator to the right of the test zero.<TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >1</TD><TD >|</TD><TD >3</TD><TD >-1</TD><TD >-4</TD><TD >1</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 3)\r<BR>\n" );
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document.write( "<TR><TD >1</TD><TD >|</TD><TD >3</TD><TD >-1</TD><TD >-4</TD><TD >1</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >3</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 1 by 3 and place the product (which is 3)  right underneath the second  coefficient (which is -1)\r<BR>\n" );
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document.write( "<TR><TD >1</TD><TD >|</TD><TD >3</TD><TD >-1</TD><TD >-4</TD><TD >1</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >3</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >3</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add 3 and -1 to get 2. Place the sum right underneath 3.\r<BR>\n" );
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document.write( "<TR><TD >1</TD><TD >|</TD><TD >3</TD><TD >-1</TD><TD >-4</TD><TD >1</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >3</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >3</TD><TD >2</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 1 by 2 and place the product (which is 2)  right underneath the third  coefficient (which is -4)\r<BR>\n" );
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document.write( "<TR><TD >1</TD><TD >|</TD><TD >3</TD><TD >-1</TD><TD >-4</TD><TD >1</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >3</TD><TD >2</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >3</TD><TD >2</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add 2 and -4 to get -2. Place the sum right underneath 2.\r<BR>\n" );
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document.write( "<TR><TD >1</TD><TD >|</TD><TD >3</TD><TD >-1</TD><TD >-4</TD><TD >1</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >3</TD><TD >2</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >3</TD><TD >2</TD><TD >-2</TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 1 by -2 and place the product (which is -2)  right underneath the fourth  coefficient (which is 1)\r<BR>\n" );
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document.write( "<TR><TD >1</TD><TD >|</TD><TD >3</TD><TD >-1</TD><TD >-4</TD><TD >1</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >3</TD><TD >2</TD><TD >-2</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >3</TD><TD >2</TD><TD >-2</TD><TD ></TD></TR>\n" );
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document.write( "    Add -2 and 1 to get -1. Place the sum right underneath -2.\r<BR>\n" );
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document.write( "<TR><TD >1</TD><TD >|</TD><TD >3</TD><TD >-1</TD><TD >-4</TD><TD >1</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >3</TD><TD >2</TD><TD >-2</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >3</TD><TD >2</TD><TD >-2</TD><TD >-1</TD></TR>\n" );
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document.write( "Since the last column adds to -1, we have a remainder of -1. This means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-1 ALIGN=MIDDLE  ALT=\"x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <b>not</b> a factor of  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E3+-+x%5E2+-+4x+%2B+1 ALIGN=MIDDLE  ALT=\"3x%5E3+-+x%5E2+-+4x+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now lets look at the bottom row of coefficients:\r<BR>\n" );
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document.write( "The first 3 coefficients (3,2,-2) form the quotient\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2+%2B+2x+-+2 ALIGN=MIDDLE  ALT=\"3x%5E2+%2B+2x+-+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and the last coefficient -1, is the remainder, which is placed over <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-1 ALIGN=MIDDLE  ALT=\"x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > like this\r<BR>\n" );
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document.write( "Putting this altogether, we get:\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%5E3+-+x%5E2+-+4x+%2B+1%29%2F%28x-1%29=3x%5E2+%2B+2x+-+2%2B-1%2F%28x-1%29 ALIGN=MIDDLE  ALT=\"%283x%5E3+-+x%5E2+-+4x+%2B+1%29%2F%28x-1%29=3x%5E2+%2B+2x+-+2%2B-1%2F%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18x%2Bx%5E2-11x ALIGN=MIDDLE  ALT=\"18x%2Bx%5E2-11x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B7x ALIGN=MIDDLE  ALT=\"x%5E2%2B7x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28x%2B7%29 ALIGN=MIDDLE  ALT=\"x%28x%2B7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the GCF <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18x%2Bx%5E2-11x ALIGN=MIDDLE  ALT=\"18x%2Bx%5E2-11x\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28x%2B7%29 ALIGN=MIDDLE  ALT=\"x%28x%2B7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "# 3\r<BR>\n" );
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document.write( "Looking at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2%2Bx-2 ALIGN=MIDDLE  ALT=\"6x%5E2%2Bx-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can see that the first term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2 ALIGN=MIDDLE  ALT=\"6x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the last term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the coefficients are 6 and -2 respectively.\r<BR>\n" );
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document.write( "Now multiply the first coefficient 6 and the last coefficient -2 to get -12. Now what two numbers multiply to -12 and add to the  middle coefficient 1? Let's list all of the factors of -12:\r<BR>\n" );
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document.write( "Factors of -12:\r<BR>\n" );
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document.write( "1,2,3,4,6,12\r<BR>\n" );
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document.write( "-1,-2,-3,-4,-6,-12 ...List the negative factors as well. This will allow us to find all possible combinations\r<BR>\n" );
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document.write( "These factors pair up and multiply to -12\r<BR>\n" );
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document.write( "(1)*(-12)\r<BR>\n" );
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document.write( "(2)*(-6)\r<BR>\n" );
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document.write( "(3)*(-4)\r<BR>\n" );
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document.write( "(-1)*(12)\r<BR>\n" );
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document.write( "(-2)*(6)\r<BR>\n" );
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document.write( "(-3)*(4)\r<BR>\n" );
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document.write( "note: remember, the product of a negative and a positive number is a negative number\r<BR>\n" );
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document.write( "Now which of these pairs add to 1? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 1\r<BR>\n" );
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document.write( "<th>First Number</th><th>Second Number</th><th>Sum</th><tr><TD  align=\"center\">1</TD><TD  align=\"center\">-12</TD><TD >1+(-12)=-11</TD></tr><tr><TD  align=\"center\">2</TD><TD  align=\"center\">-6</TD><TD >2+(-6)=-4</TD></tr><tr><TD  align=\"center\">3</TD><TD  align=\"center\">-4</TD><TD >3+(-4)=-1</TD></tr><tr><TD  align=\"center\">-1</TD><TD  align=\"center\">12</TD><TD >-1+12=11</TD></tr><tr><TD  align=\"center\">-2</TD><TD  align=\"center\">6</TD><TD >-2+6=4</TD></tr><tr><TD  align=\"center\">-3</TD><TD  align=\"center\">4</TD><TD >-3+4=1</TD></tr>\n" );
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document.write( "From this list we can see that -3 and 4 add up to 1 and multiply to -12\r<BR>\n" );
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document.write( "Now looking at the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2%2Bx-2 ALIGN=MIDDLE  ALT=\"6x%5E2%2Bx-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3x%2B4x ALIGN=MIDDLE  ALT=\"-3x%2B4x\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3x%2B4x ALIGN=MIDDLE  ALT=\"-3x%2B4x\"   DISABLED_style= \"max-width:90%; height:auto;\" > adds up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So it is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "Now let's factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2-3x%2B4x-2 ALIGN=MIDDLE  ALT=\"6x%5E2-3x%2B4x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > by grouping:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286x%5E2-3x%29%2B%284x-2%29 ALIGN=MIDDLE  ALT=\"%286x%5E2-3x%29%2B%284x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Group like terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%282x-1%29%2B2%282x-1%29 ALIGN=MIDDLE  ALT=\"3x%282x-1%29%2B2%282x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the GCF of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x ALIGN=MIDDLE  ALT=\"3x\"   DISABLED_style= \"max-width:90%; height:auto;\" > out of the first group. Factor out the GCF of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > out of the second group\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%2B2%29%282x-1%29 ALIGN=MIDDLE  ALT=\"%283x%2B2%29%282x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Since we have a common term of <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;\" >, we can combine like terms\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2-3x%2B4x-2 ALIGN=MIDDLE  ALT=\"6x%5E2-3x%2B4x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%2B2%29%282x-1%29 ALIGN=MIDDLE  ALT=\"%283x%2B2%29%282x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So this also means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2%2Bx-2 ALIGN=MIDDLE  ALT=\"6x%5E2%2Bx-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%2B2%29%282x-1%29 ALIGN=MIDDLE  ALT=\"%283x%2B2%29%282x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2%2Bx-2 ALIGN=MIDDLE  ALT=\"6x%5E2%2Bx-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2-3x%2B4x-2 ALIGN=MIDDLE  ALT=\"6x%5E2-3x%2B4x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "     Answer:\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E2%2Bx-2 ALIGN=MIDDLE  ALT=\"6x%5E2%2Bx-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%2B2%29%282x-1%29 ALIGN=MIDDLE  ALT=\"%283x%2B2%29%282x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "# 4\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E3%2B31x%5E2%2B20x ALIGN=MIDDLE  ALT=\"12x%5E3%2B31x%5E2%2B20x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2812x%5E2%2B31x%2B20%29 ALIGN=MIDDLE  ALT=\"x%2812x%5E2%2B31x%2B20%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the GCF <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now let's focus on the inner expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E2%2B31x%2B20 ALIGN=MIDDLE  ALT=\"12x%5E2%2B31x%2B20\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Looking at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E2%2B31x%2B20 ALIGN=MIDDLE  ALT=\"12x%5E2%2B31x%2B20\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can see that the first term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E2 ALIGN=MIDDLE  ALT=\"12x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the last term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20 ALIGN=MIDDLE  ALT=\"20\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the coefficients are 12 and 20 respectively.\r<BR>\n" );
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document.write( "Now multiply the first coefficient 12 and the last coefficient 20 to get 240. Now what two numbers multiply to 240 and add to the  middle coefficient 31? Let's list all of the factors of 240:\r<BR>\n" );
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document.write( "Factors of 240:\r<BR>\n" );
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document.write( "1,2,3,4,5,6,8,10,12,15,16,20,24,30,40,48,60,80,120,240\r<BR>\n" );
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document.write( "-1,-2,-3,-4,-5,-6,-8,-10,-12,-15,-16,-20,-24,-30,-40,-48,-60,-80,-120,-240 ...List the negative factors as well. This will allow us to find all possible combinations\r<BR>\n" );
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document.write( "These factors pair up and multiply to 240\r<BR>\n" );
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document.write( "1*240\r<BR>\n" );
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document.write( "2*120\r<BR>\n" );
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document.write( "3*80\r<BR>\n" );
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document.write( "4*60\r<BR>\n" );
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document.write( "5*48\r<BR>\n" );
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document.write( "6*40\r<BR>\n" );
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document.write( "8*30\r<BR>\n" );
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document.write( "10*24\r<BR>\n" );
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document.write( "12*20\r<BR>\n" );
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document.write( "15*16\r<BR>\n" );
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document.write( "(-1)*(-240)\r<BR>\n" );
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document.write( "(-2)*(-120)\r<BR>\n" );
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document.write( "(-3)*(-80)\r<BR>\n" );
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document.write( "(-4)*(-60)\r<BR>\n" );
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document.write( "(-5)*(-48)\r<BR>\n" );
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document.write( "(-6)*(-40)\r<BR>\n" );
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document.write( "(-8)*(-30)\r<BR>\n" );
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document.write( "(-10)*(-24)\r<BR>\n" );
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document.write( "(-12)*(-20)\r<BR>\n" );
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document.write( "(-15)*(-16)\r<BR>\n" );
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document.write( "note: remember two negative numbers multiplied together make a positive number\r<BR>\n" );
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document.write( "Now which of these pairs add to 31? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 31\r<BR>\n" );
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document.write( "<TABLE  border=\"1\">\n" );
document.write( "<th>First Number</th><th>Second Number</th><th>Sum</th><tr><TD  align=\"center\">1</TD><TD  align=\"center\">240</TD><TD >1+240=241</TD></tr><tr><TD  align=\"center\">2</TD><TD  align=\"center\">120</TD><TD >2+120=122</TD></tr><tr><TD  align=\"center\">3</TD><TD  align=\"center\">80</TD><TD >3+80=83</TD></tr><tr><TD  align=\"center\">4</TD><TD  align=\"center\">60</TD><TD >4+60=64</TD></tr><tr><TD  align=\"center\">5</TD><TD  align=\"center\">48</TD><TD >5+48=53</TD></tr><tr><TD  align=\"center\">6</TD><TD  align=\"center\">40</TD><TD >6+40=46</TD></tr><tr><TD  align=\"center\">8</TD><TD  align=\"center\">30</TD><TD >8+30=38</TD></tr><tr><TD  align=\"center\">10</TD><TD  align=\"center\">24</TD><TD >10+24=34</TD></tr><tr><TD  align=\"center\">12</TD><TD  align=\"center\">20</TD><TD >12+20=32</TD></tr><tr><TD  align=\"center\">15</TD><TD  align=\"center\">16</TD><TD >15+16=31</TD></tr><tr><TD  align=\"center\">-1</TD><TD  align=\"center\">-240</TD><TD >-1+(-240)=-241</TD></tr><tr><TD  align=\"center\">-2</TD><TD  align=\"center\">-120</TD><TD >-2+(-120)=-122</TD></tr><tr><TD  align=\"center\">-3</TD><TD  align=\"center\">-80</TD><TD >-3+(-80)=-83</TD></tr><tr><TD  align=\"center\">-4</TD><TD  align=\"center\">-60</TD><TD >-4+(-60)=-64</TD></tr><tr><TD  align=\"center\">-5</TD><TD  align=\"center\">-48</TD><TD >-5+(-48)=-53</TD></tr><tr><TD  align=\"center\">-6</TD><TD  align=\"center\">-40</TD><TD >-6+(-40)=-46</TD></tr><tr><TD  align=\"center\">-8</TD><TD  align=\"center\">-30</TD><TD >-8+(-30)=-38</TD></tr><tr><TD  align=\"center\">-10</TD><TD  align=\"center\">-24</TD><TD >-10+(-24)=-34</TD></tr><tr><TD  align=\"center\">-12</TD><TD  align=\"center\">-20</TD><TD >-12+(-20)=-32</TD></tr><tr><TD  align=\"center\">-15</TD><TD  align=\"center\">-16</TD><TD >-15+(-16)=-31</TD></tr>\n" );
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document.write( "From this list we can see that 15 and 16 add up to 31 and multiply to 240\r<BR>\n" );
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document.write( "Now looking at the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E2%2B31x%2B20 ALIGN=MIDDLE  ALT=\"12x%5E2%2B31x%2B20\"   DISABLED_style= \"max-width:90%; height:auto;\" >, replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=31x ALIGN=MIDDLE  ALT=\"31x\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15x%2B16x ALIGN=MIDDLE  ALT=\"15x%2B16x\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15x%2B16x ALIGN=MIDDLE  ALT=\"15x%2B16x\"   DISABLED_style= \"max-width:90%; height:auto;\" > adds up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=31x ALIGN=MIDDLE  ALT=\"31x\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So it is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=31x ALIGN=MIDDLE  ALT=\"31x\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E2%2Bhighlight%2815x%2B16x%29%2B20 ALIGN=MIDDLE  ALT=\"12x%5E2%2Bhighlight%2815x%2B16x%29%2B20\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now let's factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E2%2B15x%2B16x%2B20 ALIGN=MIDDLE  ALT=\"12x%5E2%2B15x%2B16x%2B20\"   DISABLED_style= \"max-width:90%; height:auto;\" > by grouping:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2812x%5E2%2B15x%29%2B%2816x%2B20%29 ALIGN=MIDDLE  ALT=\"%2812x%5E2%2B15x%29%2B%2816x%2B20%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Group like terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%284x%2B5%29%2B4%284x%2B5%29 ALIGN=MIDDLE  ALT=\"3x%284x%2B5%29%2B4%284x%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the GCF of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x ALIGN=MIDDLE  ALT=\"3x\"   DISABLED_style= \"max-width:90%; height:auto;\" > out of the first group. Factor out the GCF of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > out of the second group\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%2B4%29%284x%2B5%29 ALIGN=MIDDLE  ALT=\"%283x%2B4%29%284x%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Since we have a common term of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%2B5 ALIGN=MIDDLE  ALT=\"4x%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can combine like terms\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E2%2B15x%2B16x%2B20 ALIGN=MIDDLE  ALT=\"12x%5E2%2B15x%2B16x%2B20\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%2B4%29%284x%2B5%29 ALIGN=MIDDLE  ALT=\"%283x%2B4%29%284x%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So this also means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E2%2B31x%2B20 ALIGN=MIDDLE  ALT=\"12x%5E2%2B31x%2B20\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%2B4%29%284x%2B5%29 ALIGN=MIDDLE  ALT=\"%283x%2B4%29%284x%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E2%2B31x%2B20 ALIGN=MIDDLE  ALT=\"12x%5E2%2B31x%2B20\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E2%2B15x%2B16x%2B20 ALIGN=MIDDLE  ALT=\"12x%5E2%2B15x%2B16x%2B20\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "So our expression goes from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2812x%5E2%2B31x%2B20%29 ALIGN=MIDDLE  ALT=\"x%2812x%5E2%2B31x%2B20%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and factors further to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%283x%2B4%29%284x%2B5%29 ALIGN=MIDDLE  ALT=\"x%283x%2B4%29%284x%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E3%2B31x%5E2%2B20x ALIGN=MIDDLE  ALT=\"12x%5E3%2B31x%5E2%2B20x\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%283x%2B4%29%284x%2B5%29 ALIGN=MIDDLE  ALT=\"x%283x%2B4%29%284x%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "     </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );