document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197930>Question 1197930</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Construct a polynomial function with the stated properties. Reduce all fractions to lowest terms.\r<BR>\n" );
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document.write( "Third-degree, with zeros of −3, −1, and 4, and a y-intercept of −7. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #831393 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=831393\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "If k is a root of a polynomial, then x-k is a factor of that polynomial.\r<BR>\n" );
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document.write( "The roots -3, -1, 4 lead to factors x-(-3), x-(-1), and x-4<BR>\n" );
document.write( "Simplifying the factors leads to (x+3), (x+1), (x-4)\r<BR>\n" );
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document.write( "So far we have<BR>\n" );
document.write( "y = (x+3)(x+1)(x-4)\r<BR>\n" );
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document.write( "Stick an 'a' out front to be the leading coefficient<BR>\n" );
document.write( "y = a(x+3)(x+1)(x-4)\r<BR>\n" );
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document.write( "The y-intercept is -7<BR>\n" );
document.write( "Meaning x = 0 leads to y = -7<BR>\n" );
document.write( "We'll use these values to find 'a'<BR>\n" );
document.write( "y = a(x+3)(x+1)(x-4)<BR>\n" );
document.write( "-7 = a(0+3)(0+1)(0-4)<BR>\n" );
document.write( "-7 = a(3)(1)(-4)<BR>\n" );
document.write( "-7 = -12a<BR>\n" );
document.write( "a = -7/(-12)<BR>\n" );
document.write( "a = 7/12\r<BR>\n" );
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document.write( "We go from<BR>\n" );
document.write( "y = a(x+3)(x+1)(x-4)<BR>\n" );
document.write( "to<BR>\n" );
document.write( "y = (7/12)(x+3)(x+1)(x-4)\r<BR>\n" );
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document.write( "Then let's expand that out<BR>\n" );
document.write( "y = (7/12)(x+3)(x+1)(x-4)<BR>\n" );
document.write( "y = (7/12)(x+3)(x^2-4x+1x-4)<BR>\n" );
document.write( "y = (7/12)(x+3)(x^2-3x-4)<BR>\n" );
document.write( "y = (7/12)( x(x^2-3x-4)+3(x^2-3x-4) )<BR>\n" );
document.write( "y = (7/12)( x^3-3x^2-4x+3x^2-9x-12 )<BR>\n" );
document.write( "y = (7/12)( x^3-13x-12 )<BR>\n" );
document.write( "y = (7/12)x^3+(7/12)(-13x)+(7/12)(-12 )<BR>\n" );
document.write( "y = (7/12)x^3-(91/12)x-7\r<BR>\n" );
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document.write( "You can use a graphing tool like Desmos or GeoGebra to verify the answer.<BR>\n" );
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