document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=199774>Question 199774</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello!\r<BR>\n" );
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document.write( "Find all real zeros of the polynomial. Use the quadratic formula if necessary\r<BR>\n" );
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document.write( "P(x) = x^4 + x^3 - 5x^2 - 4x + 4 \r<BR>\n" );
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document.write( "thanks for your help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #150142 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=150142\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First, let's find the possible rational zeros of P(x):\r<BR>\n" );
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document.write( "Any rational zero can be found through this equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE Roots=\frac{p}{q}\"> where p and q are the factors of the last and first coefficients\r<BR>\n" );
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document.write( "So let's list the factors of 4 (the last coefficient):\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE p=\pm1, \pm2, \pm4\">\r<BR>\n" );
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document.write( "Now let's list the factors of 1 (the first coefficient):\r<BR>\n" );
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document.write( "Now let's divide each factor of the last coefficient by each factor of the first coefficient\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{1}{1}, \frac{2}{1}, \frac{4}{1}, \frac{-1}{1}, \frac{-2}{1}, \frac{-4}{1}\">\r<BR>\n" );
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document.write( "Now simplify\r<BR>\n" );
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document.write( "These are all the distinct rational zeros of the function that could occur\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE  1, 2, 4, -1, -2, -4\">\r<BR>\n" );
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document.write( "Now let's see which possible roots are actually roots.\r<BR>\n" );
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document.write( "Let's see if the possible zero <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is really a root for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4%2Bx%5E3-5x%5E2-4x%2B4 ALIGN=MIDDLE  ALT=\"x%5E4%2Bx%5E3-5x%5E2-4x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So let's make the synthetic division table for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4%2Bx%5E3-5x%5E2-4x%2B4 ALIGN=MIDDLE  ALT=\"x%5E4%2Bx%5E3-5x%5E2-4x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > given the possible zero <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "<tr><TD >1</TD><TD >|</TD><TD >1</TD><TD >1</TD><TD >-5</TD><TD >-4</TD><TD >4</TD></tr><tr><TD ></TD><TD >|</TD><TD > </TD><TD >1</TD><TD >2</TD><TD >-3</TD><TD >-7</TD></tr><tr><TD ></TD><TD ></TD><TD >1</TD><TD >2</TD><TD >-3</TD><TD >-7</TD><TD >-3</TD></tr></tr>\n" );
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document.write( "Since the remainder <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the right most entry in the last row) is <font size=4><b>not</b></font> equal to zero, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <font size=4><b>not</b></font> a zero of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4%2Bx%5E3-5x%5E2-4x%2B4 ALIGN=MIDDLE  ALT=\"x%5E4%2Bx%5E3-5x%5E2-4x%2B4\"   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( "Let's see if the possible zero <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is really a root for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4%2Bx%5E3-5x%5E2-4x%2B4 ALIGN=MIDDLE  ALT=\"x%5E4%2Bx%5E3-5x%5E2-4x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So let's make the synthetic division table for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4%2Bx%5E3-5x%5E2-4x%2B4 ALIGN=MIDDLE  ALT=\"x%5E4%2Bx%5E3-5x%5E2-4x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > given the possible zero <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "<tr><TD >2</TD><TD >|</TD><TD >1</TD><TD >1</TD><TD >-5</TD><TD >-4</TD><TD >4</TD></tr><tr><TD ></TD><TD >|</TD><TD > </TD><TD >2</TD><TD >6</TD><TD >2</TD><TD >-4</TD></tr><tr><TD ></TD><TD ></TD><TD >1</TD><TD >3</TD><TD >1</TD><TD >-2</TD><TD >0</TD></tr></tr>\n" );
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document.write( "Since the remainder <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the right most entry in the last row) is equal to zero, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a zero of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4%2Bx%5E3-5x%5E2-4x%2B4 ALIGN=MIDDLE  ALT=\"x%5E4%2Bx%5E3-5x%5E2-4x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4%2Bx%5E3-5x%5E2-4x%2B4=%28x-2%29%28x%5E3%2B3x%5E2%2Bx-2%29 ALIGN=MIDDLE  ALT=\"x%5E4%2Bx%5E3-5x%5E2-4x%2B4=%28x-2%29%28x%5E3%2B3x%5E2%2Bx-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Note: the term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2B3x%5E2%2Bx-2 ALIGN=MIDDLE  ALT=\"x%5E3%2B3x%5E2%2Bx-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > was formed by the first four values in the bottom row.\r<BR>\n" );
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document.write( "Now that you have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2B3x%5E2%2Bx-2 ALIGN=MIDDLE  ALT=\"x%5E3%2B3x%5E2%2Bx-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, you simply find the possible rational zeros for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2B3x%5E2%2Bx-2 ALIGN=MIDDLE  ALT=\"x%5E3%2B3x%5E2%2Bx-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and test to see which ones are really zeros (ie repeat the first two steps).\r<BR>\n" );
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document.write( "It turns out that the possible roots for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2B3x%5E2%2Bx-2 ALIGN=MIDDLE  ALT=\"x%5E3%2B3x%5E2%2Bx-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > are: 1, 2, -1, -2\r<BR>\n" );
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document.write( "and that -2 is a root of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2B3x%5E2%2Bx-2 ALIGN=MIDDLE  ALT=\"x%5E3%2B3x%5E2%2Bx-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Here's the synthetic division to prove it:\r<BR>\n" );
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document.write( "<tr><TD >-2</TD><TD >|</TD><TD >1</TD><TD >3</TD><TD >1</TD><TD >-2</TD></tr><tr><TD ></TD><TD >|</TD><TD > </TD><TD >-2</TD><TD >-2</TD><TD >2</TD></tr><tr><TD ></TD><TD ></TD><TD >1</TD><TD >1</TD><TD >-1</TD><TD >0</TD></tr></tr>\n" );
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document.write( "Looking at the bottom row of values (everything but the remainder), we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Bx-1 ALIGN=MIDDLE  ALT=\"x%5E2%2Bx-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So this means that\r<BR>\n" );
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document.write( "Note: this consequently means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4%2Bx%5E3-5x%5E2-4x%2B4=%28x-2%29%28x%2B2%29%28x%5E2%2Bx-1%29 ALIGN=MIDDLE  ALT=\"x%5E4%2Bx%5E3-5x%5E2-4x%2B4=%28x-2%29%28x%2B2%29%28x%5E2%2Bx-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now we just need to solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Bx-1=0 ALIGN=MIDDLE  ALT=\"x%5E2%2Bx-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > to find the last remaining zeros.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Bx-1=0 ALIGN=MIDDLE  ALT=\"x%5E2%2Bx-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation.\r<BR>\n" );
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document.write( "Notice we have a quadratic in the form of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Ax%5E2%2BBx%2BC ALIGN=MIDDLE  ALT=\"Ax%5E2%2BBx%2BC\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <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=B=1 ALIGN=MIDDLE  ALT=\"B=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=-1 ALIGN=MIDDLE  ALT=\"C=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's use the quadratic formula to solve for \"x\":\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-1+%2B-+sqrt%28+5+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"x+=+%28-1+%2B-+sqrt%28+5+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-1%2Bsqrt%285%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%28-1%2Bsqrt%285%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-1-sqrt%285%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%28-1-sqrt%285%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the expression.  \r<BR>\n" );
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document.write( "So the last two roots are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-1%2Bsqrt%285%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%28-1%2Bsqrt%285%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-1-sqrt%285%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%28-1-sqrt%285%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the four zeros of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29+=+x%5E4+%2B+x%5E3+-+5x%5E2+-+4x+%2B+4+ ALIGN=MIDDLE  ALT=\"P%28x%29+=+x%5E4+%2B+x%5E3+-+5x%5E2+-+4x+%2B+4+\"   DISABLED_style= \"max-width:90%; height:auto;\" >are:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2 ALIGN=MIDDLE  ALT=\"x=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-1%2Bsqrt%285%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%28-1%2Bsqrt%285%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-1-sqrt%285%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%28-1-sqrt%285%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Note: if you wanted to, you could compactly write the zeros as:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%22%22%2B-+2 ALIGN=MIDDLE  ALT=\"x=%22%22%2B-+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-1%2B-sqrt%285%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%28-1%2B-sqrt%285%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "just remember that there are 4 zeros.\r<BR>\n" );
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document.write( "If you have any questions, email me at <a  rel=nofollow HREF=\"mailto:jim_thompson5910@hotmail.com\">jim_thompson5910@hotmail.com</a>.<BR>\n" );
document.write( "Check out my <a  rel=nofollow HREF=\"http://www.freewebs.com/jimthompson5910/home.html\">website</a> if you are interested in tutoring. </SPAN>\n" );
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