document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1183613>Question 1183613</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The expression ax^3 + bx^2 - 5x + 2a is exactly divisible by x^2 - 3x - 4. Calculate the value of a and of b and factorise the expression completely. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #814032 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13216)</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=814032\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "And here is a third and very different way to solve the problem....<br><BR>\n" );
document.write( "We need to have<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28px%2Bq%29%28x%5E2-3x-4%29=ax%5E3%2Bbx%5E2-5x%2B2a ALIGN=MIDDLE  ALT=\"%28px%2Bq%29%28x%5E2-3x-4%29=ax%5E3%2Bbx%5E2-5x%2B2a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Perform the multiplication on the left...<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=px%5E3%2B%28q-3p%29x%5E2%2B%28-4p-3q%29x%2B%28-4q%29 ALIGN=MIDDLE  ALT=\"px%5E3%2B%28q-3p%29x%5E2%2B%28-4p-3q%29x%2B%28-4q%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "... and equate coefficients:<br><BR>\n" );
document.write( "x^3: p=a [1]<BR>\n" );
document.write( "x^2: q-3p=b [2]<BR>\n" );
document.write( "x: -4p-3q=-5 [3]<BR>\n" );
document.write( "constant: -4q=2a --> q=-0.5a [4]<br><BR>\n" );
document.write( "We have 4 equations in 4 unknowns, which could have led to a messy solution.  But things work out nicely....<br><BR>\n" );
document.write( "Substitute  [1] and [4] into [3]:<BR>\n" );
document.write( "-4a+1.5a=-5<BR>\n" );
document.write( "-2.5a=-5<BR>\n" );
document.write( "a=2<br><BR>\n" );
document.write( "Then<BR>\n" );
document.write( "p=a=2<BR>\n" );
document.write( "q=-0.5a = -1<BR>\n" );
document.write( "b = q-3p=-1-6=-7<br><BR>\n" );
document.write( "ANSWER: a=2; b=-7 (and p=2; q=-1)<br><BR>\n" );
document.write( "CHECK:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x-1%29%28x%5E2-3x-4%29=2x%5E3-7x%5E2-5x%2B4=ax%5E3%2Bbx%5E2-5x%5E2%2B2a%29 ALIGN=MIDDLE  ALT=\"%282x-1%29%28x%5E2-3x-4%29=2x%5E3-7x%5E2-5x%2B4=ax%5E3%2Bbx%5E2-5x%5E2%2B2a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );