document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197172>Question 1197172</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find a polynomial P(x) with real coefficients having a degree 6, leading coefficient 2, and zeros 3, 0 (multiplicity 3), and 4 - 4i. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #830320 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=830320\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Since P(x) has all real coefficients, this means any complex roots of the form a+bi automatically have a conjugate paired root of a-bi<BR>\n" );
document.write( "More specifically for this case we have the root 4-4i pair up with 4+4i\r<BR>\n" );
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document.write( "Let's start with x = 4+4i and do a bit of algebra like so<BR>\n" );
document.write( "x = 4+4i<BR>\n" );
document.write( "x-4 = 4i<BR>\n" );
document.write( "(x-4)^2 = (4i)^2<BR>\n" );
document.write( "x^2-8x+16 = 16i^2<BR>\n" );
document.write( "x^2-8x+16 = 16(-1)<BR>\n" );
document.write( "x^2-8x+16 = -16<BR>\n" );
document.write( "x^2-8x+16+16 = 0<BR>\n" );
document.write( "x^2-8x+32 = 0\r<BR>\n" );
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document.write( "If you were to solve x^2-8x+32 = 0 using the quadratic formula, then you'd get the two roots x = 4+4i and x = 4-4i.<BR>\n" );
document.write( "I'll let you do these verification steps.\r<BR>\n" );
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document.write( "The summary so far is that if 4-4i is a root of P(x), then it leads to x^2-8x+32 being a factor of P(x)\r<BR>\n" );
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document.write( "We also know that x = 3 is a root, so (x-3) is another factor of P(x)<BR>\n" );
document.write( "x = 0 is a root with multiplicity 3, which tells us x^3 is a factor \r<BR>\n" );
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document.write( "We have<BR>\n" );
document.write( "P(x) = 2x^3(x-3)(x^2-8x+32)\r<BR>\n" );
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document.write( "Distribute and expand everything out<BR>\n" );
document.write( "P(x) = 2x^3(x-3)(x^2-8x+32)<BR>\n" );
document.write( "P(x) = (2x^4-6x^3)(x^2-8x+32)<BR>\n" );
document.write( "P(x) = 2x^4(x^2-8x+32)-6x^3(x^2-8x+32)<BR>\n" );
document.write( "P(x) = 2x^4*x^2+2x^4*(-8x)+2x^4*(32)-6x^3(x^2)-6x^3(-8x)-6x^3(32)<BR>\n" );
document.write( "P(x) = 2x^6-16x^5+64x^4-6x^5+48x^4-192x^3<BR>\n" );
document.write( "P(x) = 2x^6+(-16x^5-6x^5)+(64x^4+48x^4)-192x^3<BR>\n" );
document.write( "P(x) = 2x^6-22x^5+112x^4-192x^3\r<BR>\n" );
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document.write( "You can use WolframAlpha or the CAS tool in GeoGebra to confirm that we have the correct polynomial function. <BR>\n" );
document.write( "Many other online tools are available.<BR>\n" );
document.write( "CAS = computer algebra system\r<BR>\n" );
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document.write( "You could also graph it to look for the x intercepts, but the downside is it won't show the complex roots.\r<BR>\n" );
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document.write( "Answer: <font color=red>P(x) = 2x^6 - 22x^5 + 112x^4 - 192x^3</font><BR>\n" );
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