document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=193037>Question 193037</A>: <I><SPAN STYLE=\"word-break: break-all;\"> State whether the following statements are true or false and justify your answer.\r<BR>\n" );
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document.write( "a.a(x-alpha)(x-beta) can always be expressed as a(x-h)^2+k<BR>\n" );
document.write( "b.a(x-h)^2+k can always be expressed as a(x-alpha)(x-beta). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #144915 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=144915\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a) The given statement is true, here's why...\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a(x-\alpha)(x-\beta)\"> ... Start with the given expression.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a(x^2-\alpha x-\beta x +\alpha\beta)\"> ... FOIL\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a(x^2-(\alpha+\beta) x +\alpha\beta)\"> ... Combine like terms.\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE b=-(\alpha+\beta)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE c=\alpha\beta\"> to get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a(x^2+bx+c)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a\left(x^2+bx+\frac{b^2}{4}-\frac{b^2}{4}+c\right)\"> ... Take half of \"b\" and square it to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2%2F4 ALIGN=MIDDLE  ALT=\"b%5E2%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Add AND subtract this inside the parenthesis.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a\left(\left(x+\frac{b}{2}\right)^2-\frac{b^2}{4}+c\right)\"> ... Factor the first three terms in the parenthesis\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a\left(x+\frac{b}{2}\right)^2+a\left(-\frac{b^2}{4}+c\right)\"> ... Distribute\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE k=a\left(-\frac{b^2}{4}+c\right)\"> to get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a\left(x+\frac{b}{2}\right)^2+k\">\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE h=-\frac{b}{2}\"> to get \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a\left(x-h\right)^2+k\">\r<BR>\n" );
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document.write( "So for ANY expression of the form <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a(x-\alpha)(x-\beta)\"> you can rewrite it as <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a\left(x-h\right)^2+k\">\r<BR>\n" );
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document.write( "============================================================\r<BR>\n" );
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document.write( "b) The given statement is false (if you are only restricted to factor over the reals)\r<BR>\n" );
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document.write( "Here's a counter-example:\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=1 ALIGN=MIDDLE  ALT=\"h=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=3 ALIGN=MIDDLE  ALT=\"k=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So the general expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%28x-h%29%5E2%2Bk ALIGN=MIDDLE  ALT=\"a%28x-h%29%5E2%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" > becomes\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%5E2%2B3 ALIGN=MIDDLE  ALT=\"%28x-1%29%5E2%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "-----------\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%5E2%2B3 ALIGN=MIDDLE  ALT=\"%28x-1%29%5E2%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-2x%2B1%2B3 ALIGN=MIDDLE  ALT=\"x%5E2-2x%2B1%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > FOIL\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-2x%2B4 ALIGN=MIDDLE  ALT=\"x%5E2-2x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "Since you CANNOT factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-2x%2B4 ALIGN=MIDDLE  ALT=\"x%5E2-2x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > over the reals, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%5E2%2B3 ALIGN=MIDDLE  ALT=\"%28x-1%29%5E2%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > CANNOT be written in the form of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a(x-\alpha)(x-\beta)\"> \r<BR>\n" );
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document.write( "Note: if you are allowed to factor over the complex numbers, then you can rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%28x-h%29%5E2%2Bk ALIGN=MIDDLE  ALT=\"a%28x-h%29%5E2%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" > into <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a(x-\alpha)(x-\beta)\"> \r<BR>\n" );
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document.write( " </SPAN>\n" );
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