document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198421>Question 1198421</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For the polynomial​ function, ​(a) list all possible rational​ zeros, ​(b) find all rational​ zeros, and (c) factor f(x).<BR>\n" );
document.write( "f(x)=x^(3)+x^(2)-37x+35 </SPAN>\n" );
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document.write( "Part (a)\r<BR>\n" );
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document.write( "Since the leading coefficient is 1, this means the possible rational roots of the polynomial is simply the factors of the last term, aka constant term (35). \r<BR>\n" );
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document.write( "Factors of 35 are:  1, 5, 7, 35\r<BR>\n" );
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document.write( "We'll also consider the negative version of each factor.\r<BR>\n" );
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document.write( "Possible rational roots: 1, -1, 5, -5, 7, -7, 35, -35\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "Use the results of part (a) to check each possible rational root to see if it's an actual rational root.<BR>\n" );
document.write( "The goal is to have f(x) = 0 for the given input x value.\r<BR>\n" );
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document.write( "Let's see if x = 1 is a root<BR>\n" );
document.write( "f(x)=x^3+x^2-37x+35<BR>\n" );
document.write( "f(1)=(1)^3+(1)^2-37(1)+35<BR>\n" );
document.write( "f(1)=0<BR>\n" );
document.write( "The result of 0 tells us that x = 1 is indeed a root or x intercept. <BR>\n" );
document.write( "This is one location where the graph either touches the x axis (and bounces away) or crosses the x axis.\r<BR>\n" );
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document.write( "Try x = -1 <BR>\n" );
document.write( "f(x)=x^3+x^2-37x+35<BR>\n" );
document.write( "f(-1)=(-1)^3+(-1)^2-37(-1)+35<BR>\n" );
document.write( "f(-1)=72<BR>\n" );
document.write( "The nonzero result tells us that x = -1 is not a root of f(x).\r<BR>\n" );
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document.write( "I'll let you try the other potential rational roots. <BR>\n" );
document.write( "There are two others.\r<BR>\n" );
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document.write( "Part (c)\r<BR>\n" );
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document.write( "The results of part (b) will play a key role in determining the factors.\r<BR>\n" );
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document.write( "For instance, we know that x = 1 is a root from the work done in part (b). <BR>\n" );
document.write( "Subtract 1 from both sides to end up with x-1 = 0 to show that (x-1) is a factor.\r<BR>\n" );
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document.write( "In general, if x = p is a root, then (x-p) is a factor.\r<BR>\n" );
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document.write( "Therefore, <BR>\n" );
document.write( "f(x) = x^3+x^2-37x+35<BR>\n" );
document.write( "f(x) = (x-1)(x-p)(x-q)\r<BR>\n" );
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document.write( "where p & q are yet to be determined. <BR>\n" );
document.write( "I'll let you find these values. <BR>\n" );
document.write( "They are the other two roots I mentioned at the end of part (b).<BR>\n" );
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