document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206674>Question 1206674</A>: <I><SPAN STYLE=\"word-break: break-all;\"> list ALL roots (rational, irrational, and/or complex) of the given polynomial equation by using the methods discussed in this lesson. SHOW YOUR WORK. \r<BR>\n" );
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document.write( " x^3 - 5x^2 + 7x - 35 = 0\r<BR>\n" );
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document.write( "Since the leading coefficient is 1, we list the factors of the last term to generate all possible rational roots.\r<BR>\n" );
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document.write( "List of possible rational roots:<BR>\n" );
document.write( "1, -1, 5, -5, 7, -7, 35, -35<BR>\n" );
document.write( "List the positive and negative versions of each.\r<BR>\n" );
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document.write( "Then plug each of them into the function to see which generates the result of 0.<BR>\n" );
document.write( "Let's try x = 1<BR>\n" );
document.write( "f(x) = x^3 - 5x^2 + 7x - 35<BR>\n" );
document.write( "f(1) = (1)^3 - 5(1)^2 + 7(1) - 35<BR>\n" );
document.write( "f(1) = -32<BR>\n" );
document.write( "The nonzero result tells us that x = 1 is not a root.<BR>\n" );
document.write( "Every other value but x = 5 will also generate nonzero results. \r<BR>\n" );
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document.write( "f(x) = x^3 - 5x^2 + 7x - 35<BR>\n" );
document.write( "f(5) = (5)^3 - 5(5)^2 + 7(5) - 35<BR>\n" );
document.write( "f(5) = 0<BR>\n" );
document.write( "Proving that x = 5 is a root and x-5 is a factor.\r<BR>\n" );
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document.write( "Let's apply synthetic division with this root.<BR>\n" );
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document.write( "<tr><TD >5</TD><TD ></TD><TD >1</TD><TD >-5</TD><TD >7</TD><TD >-35</TD></tr><tr><TD ></TD><TD ></TD><TD ></TD><TD >5</TD><TD >0</TD><TD >35</TD></tr><tr><TD ></TD><TD ></TD><TD >1</TD><TD >0</TD><TD >7</TD><TD >0</TD></tr>\n" );
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document.write( "The last item in the bottom row is 0 which is the remainder. The remainder 0 confirms that x = 5 is indeed a root.<BR>\n" );
document.write( "The other items in the bottom row form the quotient.\r<BR>\n" );
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document.write( "Quotient = 1x^2 + 0x + 7 = x^2 + 7\r<BR>\n" );
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document.write( "We determine x^3-5x^2+7x-35 = (x-5)(x^2+7)\r<BR>\n" );
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document.write( "Solving x^2+7 = 0 leads to x = i*sqrt(7) and x = -i*sqrt(7) where i = sqrt(-1)\r<BR>\n" );
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document.write( "The three roots are<BR>\n" );
document.write( "<font color=red>x = 5, x = i*sqrt(7) and x = -i*sqrt(7)</font><BR>\n" );
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