document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=484414>Question 484414</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello, I'm having trouble trying to factor each polynomial completley. The equation is 6x^3+4x^2-10x<BR>\n" );
document.write( "In words that equation would be six x cubed plus four x squared minus ten x<BR>\n" );
document.write( "Thank you so much for the help.<BR>\n" );
document.write( "-Kaitlin </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #331446 by </B><A HREF=/tutors/aboutme.mpl?userid=tinbar><B>tinbar(133)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=tinbar><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=331446\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> In the last solution I gave you, we dealt with the form of a^2-b^2, which is quick and easy to factor.\r<BR>\n" );
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document.write( "Now this poly is a little more general and requires a bit more thinking to solve, but once you understand the concept, you can use to factor all kinds of crazy polynomials.\r<BR>\n" );
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document.write( "First, we always try to factor out 'obvious' things. In 6x^3+4x^2-10x, two things are obvious: 1) Every term has a factor of 'x', so we can definitely remove that, and 2) Every term has an even coefficient, meaning we can pull out AT LEAST a factor of 2 [in other cases it's possible to pull out more, we are simply trying to take out the greatest common factor between all the terms].\r<BR>\n" );
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document.write( "So now 6x^3+4x^2-10x becomes 2x(3x^2 + 2x - 5). However, this is not completely factored yet, (3x^2 + 2x - 5) can be (possibly) factored more, let's try. \r<BR>\n" );
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document.write( "(3x^2 + 2x - 5) = (ax+b)*(cx+d), where a,b,c,d are numbers (they can be negative as well, the + signs are there just as default, they might becomes - signs). Now we have to figure out what a,b,c, and d are. We already have certain clues, for instance, a*c MUST equal 3 since when we re-expand these two brackets a*c gives us the leading term and we already know it must be 3. Similarly, we can see b*d MUST equal -5 since when we re-expand b*d gives us our last term. Now if you have been checking along as I've asked you to, you should be able to take (ax+b)*(cx+d), and expand it and be able to conclude that the middle term coefficient +2, should be given by a*d+b*c.\r<BR>\n" );
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document.write( "So we have:<BR>\n" );
document.write( "a*c=3<BR>\n" );
document.write( "b*d=-5<BR>\n" );
document.write( "a*d+b*c = 2\r<BR>\n" );
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document.write( "We will just guess since there are only few choices to satisfy these conditions, maybe a=3 and c=1, and b=-5 and d = 1, but if this was the answer, then a*d+b*c = 3*1+(-5)*1 = 3-5 = -2, so that's not the answer we are looking for, we want +2. We just have to make a slight adjustment; let a=3,c=1,b=5, and d=-1[we switched the negative sign of b to d, this is ok since we just require b*d have a final product of -5, where the '-' comes from makes no difference. With this solution we see a*d+b*c = 3*(-1)+5*1 = -3+5 = 2.\r<BR>\n" );
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document.write( "So our factored form is: 6x^3+4x^2-10x = 2x(3x^2 + 2x - 5) = 2x </SPAN>\n" );
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