document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1190851>Question 1190851</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The average speed of a car is  2x^2 + 8k - 12 km/h.  How far can it go in 3x-2  hours? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #822579 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=822579\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I'm assuming you mean to say 8x instead of 8k<BR>\n" );
document.write( "The idea is to multiply the speed with the time value to find the distance.\r<BR>\n" );
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document.write( "distance = rate*time<BR>\n" );
document.write( "distance = speed*time\r<BR>\n" );
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document.write( "So the goal is to multiply out (2x^2+8x-12)*(3x-2)\r<BR>\n" );
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document.write( "One way to multiply polynomials is to lay out a table like this<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD >2x^2</TD><TD >8x</TD><TD >-12</TD></tr><tr><TD >3x</TD><TD >A</TD><TD >B</TD><TD >C</TD></tr><tr><TD >-2</TD><TD >D</TD><TD >E</TD><TD >F</TD></tr>\n" );
document.write( "</TABLE>The letters A through F are placeholders for something else.\r<BR>\n" );
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document.write( "To figure out what that \"something else\" is, we multiply the outer pair of items.<BR>\n" );
document.write( "So the upper left corner is 6x^3 because 3x times 2x^2 = 6x^3<BR>\n" );
document.write( "This means we replace the \"A\" with 6x^3<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD >2x^2</TD><TD >8x</TD><TD >-12</TD></tr><tr><TD >3x</TD><TD >6x^3</TD><TD >B</TD><TD >C</TD></tr><tr><TD >-2</TD><TD >D</TD><TD >E</TD><TD >F</TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "The bottom right corner is 24 because -2*(-12) = 24<BR>\n" );
document.write( "Therefore, we replace the letter F placeholder with 24<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD >2x^2</TD><TD >8x</TD><TD >-12</TD></tr><tr><TD >3x</TD><TD >6x^3</TD><TD >B</TD><TD >C</TD></tr><tr><TD >-2</TD><TD >D</TD><TD >E</TD><TD >24</TD></tr>\n" );
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document.write( "The other items are handled in a very similar fashion.\r<BR>\n" );
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document.write( "I'll let you fill out the rest of the table. <BR>\n" );
document.write( "Once the table is fully complete, you add up the stuff found inside. <BR>\n" );
document.write( "If there are any like terms, make sure to combine them. \r<BR>\n" );
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document.write( "For more information, check out the box method.\r<BR>\n" );
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document.write( "A similar example is shown below.\r<BR>\n" );
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document.write( "-----------------------------\r<BR>\n" );
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document.write( "Example:\r<BR>\n" );
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document.write( "Let's say we wanted to multiply x^2+5x+6 with x-3\r<BR>\n" );
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document.write( "The starting table looks like<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD >x^2</TD><TD >5x</TD><TD >6</TD></tr><tr><TD >x</TD><TD >A</TD><TD >B</TD><TD >C</TD></tr><tr><TD >-3</TD><TD >D</TD><TD >E</TD><TD >F</TD></tr>\n" );
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document.write( "Then we have <BR>\n" );
document.write( "A = x*x^2 = x^3 in the upper left corner<BR>\n" );
document.write( "B = x*5x = 5x^2<BR>\n" );
document.write( "C = x*6 = 6x <BR>\n" );
document.write( "to finish off the first row.\r<BR>\n" );
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document.write( "For the next row, we have,<BR>\n" );
document.write( "D = -3x^2<BR>\n" );
document.write( "E = -3*5x = -15x<BR>\n" );
document.write( "F = -3*6 = -18\r<BR>\n" );
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document.write( "This is the completed table for this example<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD >x^2</TD><TD >5x</TD><TD >6</TD></tr><tr><TD >x</TD><TD >x^3</TD><TD >5x^2</TD><TD >6x</TD></tr><tr><TD >-3</TD><TD >-3x^2</TD><TD >-15x</TD><TD >-18</TD></tr>\n" );
document.write( "</TABLE>Notice that any given row is the result of distributive property. Eg: Row 1 has x*(x^2+5x+6) = x^3+5x^2+6x\r<BR>\n" );
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document.write( "Then we add up the stuff found in the table (items A through F earlier)<BR>\n" );
document.write( "A+B+C+D+E+F = x^3+5x^2+6x+(-3x^2)+(-15x)+(-18) = x^3+2x^2-9x-18\r<BR>\n" );
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document.write( "Note we added these two pairs of like terms <BR>\n" );
document.write( "5x^2 and -3x^2 to get 2x^2<BR>\n" );
document.write( "6x and -15x to get -9x\r<BR>\n" );
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document.write( "Therefore,<BR>\n" );
document.write( "(x-3)*(x^2+5x+6) = x^3+2x^2-9x-18<BR>\n" );
document.write( "This is of course an example and <u>not</u> the final answer to your particular question. However, you will follow the same type of steps.<BR>\n" );
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