document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=727713>Question 727713</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do I factor 4x^2-31x+21? Thanks!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #445185 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=445185\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> It is not easy, but <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2-31x%2B21 ALIGN=MIDDLE  ALT=\"4x%5E2-31x%2B21\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored as a product of two binomials with integer coefficients.<BR>\n" );
document.write( "Some (not all) quadratic polynomials can be factored.<BR>\n" );
document.write( "Of the ones than can be factored, some (not all) have factors with integer coefficients.<BR>\n" );
document.write( "Here's how to factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2-31x%2B21 ALIGN=MIDDLE  ALT=\"4x%5E2-31x%2B21\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( " <BR>\n" );
document.write( "STEP 1<BR>\n" );
document.write( "Multiply the leading coefficient times the independent (constant) term.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A21=84 ALIGN=MIDDLE  ALT=\"4%2A21=84\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "STEP 2<BR>\n" );
document.write( "Look for pairs of integers that multiply to yield that product, but do not worry about negative signs yet.<BR>\n" );
document.write( "There are 6 such pairs of factors.<BR>\n" );
document.write( "(You would list them the same way if the product had been -84)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2A84=84 ALIGN=MIDDLE  ALT=\"1%2A84=84\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A42=84 ALIGN=MIDDLE  ALT=\"2%2A42=84\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2A28=84 ALIGN=MIDDLE  ALT=\"3%2A28=84\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A21=84 ALIGN=MIDDLE  ALT=\"4%2A21=84\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2A14=84 ALIGN=MIDDLE  ALT=\"6%2A14=84\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2A12=84 ALIGN=MIDDLE  ALT=\"7%2A12=84\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "STEP 3<BR>\n" );
document.write( "Find the pair of factors (including negative signs as needed) whose product is the product found above, and whose sum equals the coefficient of the term in x.<BR>\n" );
document.write( "We are looking for factors whose product is 84 and whose sum is -31.<BR>\n" );
document.write( "They will both be negative, so their absolute values will add to 31.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B84=85 ALIGN=MIDDLE  ALT=\"1%2B84=85\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2B42=44 ALIGN=MIDDLE  ALT=\"2%2B42=44\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%283%2B28=31%29 ALIGN=MIDDLE  ALT=\"highlight%283%2B28=31%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2B21=25 ALIGN=MIDDLE  ALT=\"4%2B21=25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2B14=20 ALIGN=MIDDLE  ALT=\"6%2B14=20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2B12=19 ALIGN=MIDDLE  ALT=\"7%2B12=19\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The pair of factors we are looking for is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-28 ALIGN=MIDDLE  ALT=\"-28\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "STEP 4<BR>\n" );
document.write( "Write the factors as coefficients of terms in x, and re-write your polynomial with those two new terms instead of the original term in x.<BR>\n" );
document.write( "We write<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3x ALIGN=MIDDLE  ALT=\"-3x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-28x ALIGN=MIDDLE  ALT=\"-28x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and using those two expressions to 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;\" > in the polynomial to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2-31x%2B21=4x%5E2-3x-28x%2B21 ALIGN=MIDDLE  ALT=\"4x%5E2-31x%2B21=4x%5E2-3x-28x%2B21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "STEP 4<BR>\n" );
document.write( "Factor by parts, taking out common factors for pairs of the terms.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2-28x=4x%28x-7%29 ALIGN=MIDDLE  ALT=\"4x%5E2-28x=4x%28x-7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3x%2B21=-3%28x-7%29 ALIGN=MIDDLE  ALT=\"-3x%2B21=-3%28x-7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > so to start we write<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2-3x-28x%2B21=%284x%5E2-28x%29%2B%28-3x%2B21%29=4x%28x-7%29-3%28x-7%29 ALIGN=MIDDLE  ALT=\"4x%5E2-3x-28x%2B21=%284x%5E2-28x%29%2B%28-3x%2B21%29=4x%28x-7%29-3%28x-7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then we take out <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-7%29 ALIGN=MIDDLE  ALT=\"%28x-7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as common factor to finish and we have the factoring as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2-3x-28x%2B21=%284x%5E2-28x%29%2B%28-3x%2B21%29=4x%28x-7%29-3%28x-7%29=highlight%28%284x-3%29%28x-7%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );