document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=266534>Question 266534</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a polynomial in x has degree 3. The coefficient of x^2 is 3 less than the coefficient of x^3. The coefficient of x is three times the coefficient of x^2. The remaining coefficient is 2 more than the coefficient of x^3. The sum of the coefficients is -4. Find the polynomial.\r<BR>\n" );
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document.write( "Please help. I am so lost on this chapter. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #195833 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=195833\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> general form of your equation is:\r<BR>\n" );
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document.write( "f(x) = ax^3 + bx^2 + cx + d\r<BR>\n" );
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document.write( "a is the coefficient of x^3<BR>\n" );
document.write( "b is the coefficient of x^2<BR>\n" );
document.write( "c is the coefficient of x^1<BR>\n" );
document.write( "d is the coefficient of x^0 which makes it the constant term.\r<BR>\n" );
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document.write( "the problem states that:\r<BR>\n" );
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document.write( "The coefficient of x^2 is 3 less than the coefficient of x^3. <BR>\n" );
document.write( "This means that b = a-3\r<BR>\n" );
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document.write( "The coefficient of x is three times the coefficient of x^2. <BR>\n" );
document.write( "This means that c = 3*b\r<BR>\n" );
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document.write( "The remaining coefficient is 2 more than the coefficient of x^3. <BR>\n" );
document.write( "This means that d = a+2\r<BR>\n" );
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document.write( "The sum of the coefficients is -4.<BR>\n" );
document.write( "This means that a + b + c + d = -4.\r<BR>\n" );
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document.write( "the equations  you have to work with are:\r<BR>\n" );
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document.write( "b = a-3<BR>\n" );
document.write( "c = 3*b<BR>\n" );
document.write( "d = a+2<BR>\n" );
document.write( "a + b + c + d = -4\r<BR>\n" );
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document.write( "these wind up being a system of 4 equations in 4 unknowns that have to be solved simultaneously.\r<BR>\n" );
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document.write( "this system can be solved by substitution or by elimination.\r<BR>\n" );
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document.write( "we will solve by substitution.\r<BR>\n" );
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document.write( "start with your final equation of:\r<BR>\n" );
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document.write( "a + b + c + d = -4\r<BR>\n" );
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document.write( "substitute 3*b for c to get:\r<BR>\n" );
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document.write( "a + b + 3*b + d = -4\r<BR>\n" );
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document.write( "substitute a-3 for b to get:\r<BR>\n" );
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document.write( "a + a - 3 + 3*(a-3) + d = -4\r<BR>\n" );
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document.write( "substitute a+2 for d to get:\r<BR>\n" );
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document.write( "a + a - 3 + 3*(a-3) + a + 2 = -4\r<BR>\n" );
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document.write( "simplify to get:\r<BR>\n" );
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document.write( "a + a - 3 + 3*a - 9 + a + 2 = -4\r<BR>\n" );
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document.write( "combine like terms to get:\r<BR>\n" );
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document.write( "6*a -10 = -4\r<BR>\n" );
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document.write( "add 10 to both sides of this equation to get:\r<BR>\n" );
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document.write( "6*a = -4 + 10 = 6\r<BR>\n" );
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document.write( "divide both sides of this equation to get:\r<BR>\n" );
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document.write( "a = 1\r<BR>\n" );
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document.write( "your equations that you needed to solve simultaneously are:\r<BR>\n" );
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document.write( "b = a-3<BR>\n" );
document.write( "c = 3*b<BR>\n" );
document.write( "d = a+2<BR>\n" );
document.write( "a + b + c + d = -4\r<BR>\n" );
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document.write( "since a = 1, you get:\r<BR>\n" );
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document.write( "b = 1-3 = -2<BR>\n" );
document.write( "c = 3*b = 3*-2 = -6<BR>\n" );
document.write( "d = a+2 = 1+2 = 3\r<BR>\n" );
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document.write( "your coefficients become:\r<BR>\n" );
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document.write( "a = 1<BR>\n" );
document.write( "b = -2<BR>\n" );
document.write( "c = -6<BR>\n" );
document.write( "d = 3\r<BR>\n" );
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document.write( "a + b + c + d = -4 becomes:\r<BR>\n" );
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document.write( "1 - 2 - 6 + 3 = -4\r<BR>\n" );
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document.write( "this becomes:\r<BR>\n" );
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document.write( "-8 + 4 = -4 which becomes:\r<BR>\n" );
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document.write( "-4 = -4 confirming the values for a,b,c,d are good.\r<BR>\n" );
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document.write( "your equation of your polynomial becomes:\r<BR>\n" );
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document.write( "f(x) = x^3 -2x^2 -6x + 3\r<BR>\n" );
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document.write( "the x^2 coefficient is equal to 3 less than the x^3 coefficient (1-3=-2).<BR>\n" );
document.write( "the x coefficient is equal to 3 times the x^2 coefficient (3*(-2)=-6).<BR>\n" );
document.write( "the remaining coefficient is equal to 2 more than the x^3 coefficient (1+2=3).\r<BR>\n" );
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document.write( "your answer is:\r<BR>\n" );
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document.write( "f(x) = x^3 -2x^2 -6x + 3<BR>\n" );
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