document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209771>Question 1209771</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose the polynomial p(x)=x^3+ax^2+bc+c has the property that the mean of its zeroes, the product of its zeroes, and the sum of its coefficients are all equal. If the y-intercept of the graph of y=p(x) is 0, what is b? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850279 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=850279\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step-by-step:\r<BR>\n" );
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document.write( "**1. Define the Zeros:**\r<BR>\n" );
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document.write( "* Let the zeros of p(x) be r, s, and t.\r<BR>\n" );
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document.write( "**2. Relationships Between Zeros and Coefficients:**\r<BR>\n" );
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document.write( "* **Mean of Zeros:** (r + s + t) / 3 = -a / 3<BR>\n" );
document.write( "* **Product of Zeros:** rst = -c<BR>\n" );
document.write( "* **Sum of Coefficients:** p(1) = 1 + a + b + c\r<BR>\n" );
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document.write( "**3. Given Conditions:**\r<BR>\n" );
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document.write( "* The mean of the zeros, the product of the zeros, and the sum of the coefficients are all equal.<BR>\n" );
document.write( "* The y-intercept of p(x) is 0. This means p(0) = 0.\r<BR>\n" );
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document.write( "**4. Use the Y-Intercept Condition:**\r<BR>\n" );
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document.write( "* p(0) = 0^3 + a(0)^2 + b(0) + c = 0<BR>\n" );
document.write( "* c = 0\r<BR>\n" );
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document.write( "**5. Apply the Equal Condition:**\r<BR>\n" );
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document.write( "* Since c = 0, the product of the zeros is rst = -c = 0. This implies at least one of the zeros is 0. Let's say r = 0.<BR>\n" );
document.write( "* Now, we have:<BR>\n" );
document.write( "    * (r + s + t) / 3 = -a / 3<BR>\n" );
document.write( "    * rst = 0<BR>\n" );
document.write( "    * 1 + a + b + c = 1 + a + b<BR>\n" );
document.write( "* Since all three are equal:<BR>\n" );
document.write( "    * -a / 3 = 0<BR>\n" );
document.write( "    * 1 + a + b = 0\r<BR>\n" );
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document.write( "**6. Solve for a and b:**\r<BR>\n" );
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document.write( "* From -a / 3 = 0, we get a = 0.<BR>\n" );
document.write( "* Substitute a = 0 into 1 + a + b = 0:<BR>\n" );
document.write( "    * 1 + 0 + b = 0<BR>\n" );
document.write( "    * b = -1\r<BR>\n" );
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document.write( "**7. Check the Mean of Zeros:**\r<BR>\n" );
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document.write( "* Since a = 0, the mean of the zeros is -a / 3 = 0.<BR>\n" );
document.write( "* Since r = 0, we have (0 + s + t) / 3 = 0, which means s + t = 0.<BR>\n" );
document.write( "* Also, 1 + a + b = 1 + 0 - 1 = 0, which confirms the condition.\r<BR>\n" );
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document.write( "**Answer:**\r<BR>\n" );
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document.write( "The value of b is -1.<BR>\n" );
document.write( " </SPAN>\n" );
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