document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1174668>Question 1174668</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For what values of t can 10x^2+tx+8 be written as the product of two binomials?\r<BR>\n" );
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document.write( "I don't really get what we're supposed to do in this question. Please explain. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #800138 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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/cgi-bin/show-face.mpl?user=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=800138\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10x%5E2%2Btx%2B8+=+%28ax%2Bb%29%28cx%2Bd%29 ALIGN=MIDDLE  ALT=\"10x%5E2%2Btx%2B8+=+%28ax%2Bb%29%28cx%2Bd%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Since the leading coefficient and constant term are both positive, the coefficients of a and c can be either both positive or both negative; likewise for b and d.<br><BR>\n" );
document.write( "But allowing both pairs to be either positive or negative doesn't provide any different linear terms in the product.  For example...<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%2B3%29%283x-1%29+=+6x%5E2%2B7x-3 ALIGN=MIDDLE  ALT=\"%282x%2B3%29%283x-1%29+=+6x%5E2%2B7x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-2x-3%29%28-3x%2B1%29+=+6x%5E2%2B7x-3 ALIGN=MIDDLE  ALT=\"%28-2x-3%29%28-3x%2B1%29+=+6x%5E2%2B7x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So, for determining the number of possible values of the linear coefficient t, we can assume coefficients a and c are positive.  Then the possible factorizations with integer coefficients have...<BR>\n" );
document.write( "(1) (a,c) =(10,1) or (5,2) or (2,5) or (1,10); and<BR>\n" );
document.write( "(2) (b,d) = (8,1) or (4,2) or (2,4) or (1,8) OR (-8,-1) or (-4,-2) or (-2,-4) or (-1,-8)<br><BR>\n" );
document.write( "Every combination of choices from (1) and (2) will produce a different value for t.<br><BR>\n" );
document.write( "If the question asked only for the NUMBER of different possible values for t, we would know the answer at this point: 4 choices for the pair of leading coefficients and 8 choices for the pair of constants, so 4*8=32 different values of t.<br><BR>\n" );
document.write( "However, the question asks for a list of all the possible values of t; so there is still some work for you to do to finish the problem.<br><BR>\n" );
document.write( "Note that in making the complete list of the possible values of t, you can use only the positive pairs of values for the constants; whatever (positive) value of t you get, the corresponding negative value of t can be obtained using the corresponding pair of negative constants.<br><BR>\n" );
document.write( "So determine the t values using the positive choices for constants b and d to get 16 of the 32 possible values for t.  Then the remaining 16 possible values of t are the corresponding negative values.<br><BR>\n" );
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