document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209716>Question 1209716</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let a, b, c be the roots of p(x) = x^3 + 7x^2 + 10x - 13 - 5x^3 + 25x^2 + 44x - 15.  Find the value of\r<BR>\n" );
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document.write( "1/(ab + ac) + 1/(ab + bc) + 1/(ac + bc) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849933 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849933\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this problem:\r<BR>\n" );
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document.write( "1. **Simplify the polynomial:**<BR>\n" );
document.write( "   p(x) = x³ + 7x² + 10x - 13 - 5x³ + 25x² + 44x - 15<BR>\n" );
document.write( "   p(x) = -4x³ + 32x² + 54x - 28\r<BR>\n" );
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document.write( "2. **Relate roots to coefficients:**<BR>\n" );
document.write( "   For a cubic polynomial ax³ + bx² + cx + d = 0 with roots a, b, and c, we have the following relationships:<BR>\n" );
document.write( "     * a + b + c = -b/a<BR>\n" );
document.write( "     * ab + ac + bc = c/a<BR>\n" );
document.write( "     * abc = -d/a\r<BR>\n" );
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document.write( "   In our case, -4x³ + 32x² + 54x - 28 = 0. Dividing by -4 gives<BR>\n" );
document.write( "   x³ - 8x² - (27/2)x + 7 = 0<BR>\n" );
document.write( "   So,<BR>\n" );
document.write( "     * a + b + c = 8<BR>\n" );
document.write( "     * ab + ac + bc = -27/2<BR>\n" );
document.write( "     * abc = -7\r<BR>\n" );
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document.write( "3. **Simplify the expression:**<BR>\n" );
document.write( "   We want to find the value of:<BR>\n" );
document.write( "   1/(ab + ac) + 1/(ab + bc) + 1/(ac + bc)\r<BR>\n" );
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document.write( "   We can rewrite this as:<BR>\n" );
document.write( "   1/[a(b + c)] + 1/[b(a + c)] + 1/[c(a + b)]\r<BR>\n" );
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document.write( "   Find a common denominator:<BR>\n" );
document.write( "   [b(a + c)c(a + b) + a(b + c)c(a + b) + a(b + c)b(a + c)] / [a(b + c)b(a + c)c(a + b)]\r<BR>\n" );
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document.write( "   Expand the numerator:<BR>\n" );
document.write( "   [abc + bc² + abc + b²c + a²c + ac² + a²b + ab² + abc + bc² + abc + b²c] / [abc(a + b)(b + c)(c + a)]\r<BR>\n" );
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document.write( "   Simplify the numerator:<BR>\n" );
document.write( "   [2abc + bc² + b²c + a²c + ac² + a²b + ab²] / [abc(a + b)(b + c)(c + a)]\r<BR>\n" );
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document.write( "   Notice that the numerator can be rearranged as:<BR>\n" );
document.write( "   [ab(a + b) + bc(b + c) + ac(a + c) + 2abc]\r<BR>\n" );
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document.write( "4. **Substitute the relationships:**\r<BR>\n" );
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document.write( "We have:<BR>\n" );
document.write( "1/(ab + ac) + 1/(ab + bc) + 1/(ac + bc) = [ab(a+b) + bc(b+c) + ac(a+c) + 2abc]/[abc(a+b)(b+c)(c+a)]\r<BR>\n" );
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document.write( "We also know that (a+b)(b+c)(c+a) = (a+b+c)(ab+bc+ca) - abc, so (a+b)(b+c)(c+a) = 8*(-27/2) - (-7) = -108 + 7 = -101\r<BR>\n" );
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document.write( "ab(a+b) + bc(b+c) + ac(a+c) = (a+b+c)(ab+bc+ac) - 3abc = 8*(-27/2) - 3(-7) = -108 + 21 = -87\r<BR>\n" );
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document.write( "Therefore, the expression becomes:<BR>\n" );
document.write( "(-87 + 2(-7)) / (-7 * -101) = (-87 - 14) / 707 = -101/707 = -1/7\r<BR>\n" );
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document.write( "Therefore, 1/(ab + ac) + 1/(ab + bc) + 1/(ac + bc) = -1/7.<BR>\n" );
document.write( " </SPAN>\n" );
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