document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209715>Question 1209715</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let P(x) be a polynomial of the form<BR>\n" );
document.write( "P(x) = 2x^3 + ax^2 - 23x + c,<BR>\n" );
document.write( "such that 12 and 7 are roots of P(x). What is the third root?\r<BR>\n" );
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document.write( "For the polynomial in part (a), compute the ordered pair (a,c).  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849934 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=849934\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to find the third root and the values of *a* and *c*:\r<BR>\n" );
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document.write( "1. **Use the factor theorem:** Since 1/2 and 7 are roots of P(x), then (x - 1/2) and (x - 7) are factors of P(x).  Therefore, we can write P(x) as:\r<BR>\n" );
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document.write( "   P(x) = 2(x - 1/2)(x - 7)(x - r)\r<BR>\n" );
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document.write( "   where 'r' is the third root. The factor of 2 is included so that the coefficient of the x³ term matches the given polynomial.\r<BR>\n" );
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document.write( "2. **Expand the factored form:**<BR>\n" );
document.write( "   P(x) = 2(x - 1/2)(x - 7)(x - r)<BR>\n" );
document.write( "   P(x) = (2x - 1)(x - 7)(x - r)<BR>\n" );
document.write( "   P(x) = (2x² - 15x + 7)(x - r)<BR>\n" );
document.write( "   P(x) = 2x³ - 15x² + 7x - 2rx² + 15rx - 7r<BR>\n" );
document.write( "   P(x) = 2x³ + (-15 - 2r)x² + (7 + 15r)x - 7r\r<BR>\n" );
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document.write( "3. **Compare coefficients:** Now, compare the coefficients of the expanded form with the given form P(x) = 2x³ + ax² - 23x + c:\r<BR>\n" );
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document.write( "   * Coefficient of x²:  a = -15 - 2r<BR>\n" );
document.write( "   * Coefficient of x: -23 = 7 + 15r<BR>\n" );
document.write( "   * Constant term: c = -7r\r<BR>\n" );
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document.write( "4. **Solve for r:** From the coefficient of x, we can solve for r:\r<BR>\n" );
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document.write( "   -23 = 7 + 15r<BR>\n" );
document.write( "   -30 = 15r<BR>\n" );
document.write( "   r = -2\r<BR>\n" );
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document.write( "5. **Find a and c:** Now that we know r = -2, we can find *a* and *c*:\r<BR>\n" );
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document.write( "   a = -15 - 2(-2) = -15 + 4 = -11<BR>\n" );
document.write( "   c = -7(-2) = 14\r<BR>\n" );
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document.write( "Therefore, the third root is -2, and the ordered pair (a, c) is (-11, 14).<BR>\n" );
document.write( " </SPAN>\n" );
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