document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170225>Question 1170225</A>: <I><SPAN STYLE=\"word-break: break-all;\"> the binomial (x+4) is a factor of the following polynomials, find the value of k.<BR>\n" );
document.write( "1.2x^3-3x^2-kx+60<BR>\n" );
document.write( "2.7x^2+kx-12<BR>\n" );
document.write( "3.kx^3-10x^2-137x+60<BR>\n" );
document.write( "4.4x^2+kx-4<BR>\n" );
document.write( "5.8x^3+kx^2-53x+12\r<BR>\n" );
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document.write( "Find the values of A, B and C if (x-1),(x+2), and (x-2) are factors of <BR>\n" );
document.write( "2Ax^4-Bx^3-Cx-16 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #795082 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52786)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=795082\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .\r<BR>\n" );
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document.write( "In problems 1 - 5 below,  &nbsp;the idea of the solution is the same.\r<BR>\n" );
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document.write( "You substitute the value of the root  &nbsp;x= -4 &nbsp;into the formula and equate this expression to zero.\r<BR>\n" );
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document.write( "It gives you an equation to find &nbsp;\"k\".\r<BR>\n" );
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document.write( "Having this idea explained to you, &nbsp;you can complete the rest on your own.\r<BR>\n" );
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document.write( "Comment from student: what can you answer number 1 so I know what to do to others?\r<BR>\n" );
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document.write( "<U>My response</U> :\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It is &nbsp;REALLY &nbsp;GOOD &nbsp;IDEA &nbsp;start working with one specific case, &nbsp;instead of posting &nbsp;5 &nbsp;same type problems in one post.\r<BR>\n" );
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document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "1.  the binomial (x+4) is a factor of the following polynomials, find the value of k.\r\n" );
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document.write( "    2x^3 - 3x^2 - kx + 60\r\n" );
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document.write( "Since (x+4) is the root of the given polynomial, it means, due to the Remainder Theorem,\r\n" );
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document.write( "that (-4) is the root of the polynomial.\r\n" );
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document.write( "So, we substitute x= -4 into the polynomial and equate this expression to zero (since -4 is the root (!) )\r\n" );
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document.write( "We get then\r\n" );
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document.write( "    2*(-4)^3 - 3*(-4)^2 - k*(-4) + 60 = 0.\r\n" );
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document.write( "It is our BASIC EQUATION to find the unknown value of \"k\" :\r\n" );
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document.write( "   2*(-64) - 3*(16) - k*(-4) + 60 = 0\r\n" );
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document.write( "   -128    - 48     + 4k     + 60 = 0\r\n" );
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document.write( "   -116             + 4k          = 0\r\n" );
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document.write( "                      4k          = 116.\r\n" );
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document.write( "                       k          = 116/4 = 29.    \r\n" );
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document.write( "<U>ANSWER</U>.  k = 29.\r\n" );
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document.write( "Solved.\r<BR>\n" );
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document.write( "Do other cases IN THE SAME WAY.\r<BR>\n" );
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document.write( "It is a good problem, &nbsp;and it is my pleasure to teach you.\r<BR>\n" );
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document.write( "Do you understand everything in my solution ?\r<BR>\n" );
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document.write( "If you have questions, &nbsp;do not hesitate to post them to me.\r<BR>\n" );
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document.write( "Never submit many problems/questions of the same type in one post.\r<BR>\n" );
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document.write( "It is really irritates tutors; works against you and creates bad impression about a visitor.\r<BR>\n" );
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document.write( "By the way, &nbsp;it is &nbsp;PROHIBITED &nbsp;by the rules of this forum.\r<BR>\n" );
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document.write( "Happy learning &nbsp;(!)\r<BR>\n" );
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document.write( "&nbsp;&nbsp; <B>Theorem</B> &nbsp;&nbsp;(the <B><I>remainder theorem</I></B>)\r<BR>\n" );
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document.write( "&nbsp;&nbsp; <B>1</B>. The remainder of division the polynomial &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp; by the binomial &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-a ALIGN=MIDDLE  ALT=\"x-a\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp; is equal to the value &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28a%29 ALIGN=MIDDLE  ALT=\"f%28a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp; of the polynomial. \r<BR>\n" );
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document.write( "&nbsp;&nbsp; <B>2</B>. The binomial &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-a ALIGN=MIDDLE  ALT=\"x-a\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp; divides the polynomial &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp; if and only if the value of &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp; is the root of the polynomial &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;i.e. &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28a%29+=+0 ALIGN=MIDDLE  ALT=\"f%28a%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "&nbsp;&nbsp; <B>3</B>. The binomial &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-a ALIGN=MIDDLE  ALT=\"x-a\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp; factors the polynomial &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp; if and only if the value of &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >&nbsp; is the root of the polynomial &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, &nbsp;i.e. &nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28a%29+=+0 ALIGN=MIDDLE  ALT=\"f%28a%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "See the lessons\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/Divisibility-of-polynomial-f%28x%29-by-binomial-x-a.lesson>Divisibility of polynomial f(x) by binomial x-a and the Remainder theorem</A>\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/Solved-problems-on-the-Remainder-theorem.lesson>Solved problems on the Remainder thoerem</A>\r<BR>\n" );
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document.write( "in this site.\r<BR>\n" );
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document.write( "Also, &nbsp;you have this free of charge online textbook in ALGEBRA-II in this site\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;<A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.\r<BR>\n" );
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document.write( "The referred lessons are the part of this online textbook under the topic <BR>\n" );
document.write( "\"<U>Divisibility of polynomial f(x) by binomial (x-a). The Remainder theorem</U>\".\r<BR>\n" );
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document.write( "Save the link to this online textbook together with its description\r<BR>\n" );
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document.write( "Free of charge online textbook in ALGEBRA-I<BR>\n" );
document.write( "https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson\r<BR>\n" );
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document.write( "to your archive and use it when it is needed.\r<BR>\n" );
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