document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=468064>Question 468064</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 4. Use the remainder theorem to: <BR>\n" );
document.write( "(A)find the remainder when f(x) is divided by x – c, and <BR>\n" );
document.write( "(B)determine whether x – c is a factor of f(x). <BR>\n" );
document.write( "Please show all of your work.<BR>\n" );
document.write( "f(x)=x³+3x²-8x+10;x-5 \r<BR>\n" );
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document.write( "I appreciate any help, thanks. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #321152 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=321152\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> here's a good reference on remainder theorem.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://www.mathsisfun.com/algebra/polynomials-remainder-factor.html\" target = \"_blank\">http://www.mathsisfun.com/algebra/polynomials-remainder-factor.html</a>\r<BR>\n" );
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document.write( "the gist of what's in the reference is as follows:\r<BR>\n" );
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document.write( "The remainder theorem states:\r<BR>\n" );
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document.write( "When you divide a polynomial f(x) by x-c the remainder r will be f(c)\r<BR>\n" );
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document.write( "The factor theorem states:\r<BR>\n" );
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document.write( "When f(c)=0 then x-c is a factor of the polynomial\r<BR>\n" );
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document.write( "what does this mean to your problem?\r<BR>\n" );
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document.write( "your equation is:\r<BR>\n" );
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document.write( "f(x) = x^3 + 3x^2 - 8x + 10\r<BR>\n" );
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document.write( "you want to divide this by (x-5)\r<BR>\n" );
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document.write( "you want to know whether (x-5) is a factor of x^3 + 3x^2 - 8x + 10\r<BR>\n" );
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document.write( "from the remainder theorem, if (x-5) is a factor of x^3 + 3x^2 - 8x + 10, then f(5) must be equal to 0.\r<BR>\n" );
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document.write( "replacing x with 5 in the equation, you get:\r<BR>\n" );
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document.write( "(5)^3 + 3(5^2) - 8(5) + 10 = 125 + 75 - 40 + 10 = 170\r<BR>\n" );
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document.write( "since the answer is not 0, this means that (x-5) is NOT a factor of x^3 + 3x^2 - 8x + 10.\r<BR>\n" );
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document.write( "being a factor of the equation, means that (x-5) would have been a root of the equation.\r<BR>\n" );
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document.write( "if (x-5) was a root of the equation, then the graph of the equation should have shown that the value of y when x = 5 was 0.\r<BR>\n" );
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document.write( "the graph of the equation is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-10%2C10%2C-200%2C200%2Cx%5E3+%2B+3x%5E2+-+8x+%2B+10%2C170%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C600%2C-10%2C10%2C-200%2C200%2Cx%5E3+%2B+3x%5E2+-+8x+%2B+10%2C170%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "from the graph, it looks like the graph might have a root at x = -5, but definitely NOT at x = 5.\r<BR>\n" );
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document.write( "if the equation had a root at x = -5, then (x+5) would be a factor of the equation, and when you divided the equation by (x+5), you should get a remainder of 0.\r<BR>\n" );
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document.write( "f(-5) = (-5)^3 + 3*(-5)^2 - 8*(-5) + 10 = -125 + 75 + 40 + 10 = 0\r<BR>\n" );
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document.write( "since the remainder is 0, this means that (x+5) IS a factor of the equation.\r<BR>\n" );
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document.write( "unfortunately, you were not asked that.\r<BR>\n" );
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document.write( "you were asked to find if (x-5) was a factor of the equation.\r<BR>\n" );
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document.write( "it is not.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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