document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192627>Question 1192627</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find a polynomial function P of the lowest possible​ degree, having real​ coefficients, a leading coefficient of​ 1, and with the given zeros.<BR>\n" );
document.write( "2+2i, -1, and 2<BR>\n" );
document.write( "any help is very appreciated </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #824530 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=824530\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "The term \"root\" is the same as a \"zero\" of a function.\r<BR>\n" );
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document.write( "x = 2+2i is one root<BR>\n" );
document.write( "x = 2-2i is the conjugate pair to the previous root<BR>\n" );
document.write( "The roots come in conjugate pairs like this to ensure that the coefficients of P(x) are real numbers.<BR>\n" );
document.write( "The general template is that a+bi and a-bi are complex conjugate pairs.\r<BR>\n" );
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document.write( "Let's apply these steps<BR>\n" );
document.write( "x = 2+2i<BR>\n" );
document.write( "x-2 = 2i<BR>\n" );
document.write( "(x-2)^2 = (2i)^2<BR>\n" );
document.write( "(x-2)^2 = 4i^2<BR>\n" );
document.write( "(x-2)^2 = 4(-1)<BR>\n" );
document.write( "(x-2)^2 = -4<BR>\n" );
document.write( "(x-2)^2+4 = 0<BR>\n" );
document.write( "You should find that x = 2-2i also leads to the equation above.\r<BR>\n" );
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document.write( "Expand things out and simplify.<BR>\n" );
document.write( "(x-2)^2+4 = 0<BR>\n" );
document.write( "x^2-4x+4+4 = 0<BR>\n" );
document.write( "x^2-4x+8 = 0<BR>\n" );
document.write( "To verify things so far, use the quadratic formula for x^2-4x+8 and you should get the complex roots x = 2+2i and x = 2-2i.\r<BR>\n" );
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document.write( "So far we've shown that if x = 2+2i and x = 2-2i are roots of P(x), then x^2-4x+8 is a factor of P(x).\r<BR>\n" );
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document.write( "Similarly, if x = -1 is a root, then x+1 is a factor.<BR>\n" );
document.write( "And if x = 2 is a root, then x-2 is another factor.\r<BR>\n" );
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document.write( "The three factors we have are:<BR>\n" );
document.write( "x^2-4x+8<BR>\n" );
document.write( "x+1<BR>\n" );
document.write( "x-2\r<BR>\n" );
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document.write( "Multiply them out and expand like so<BR>\n" );
document.write( "(x^2-4x+8)(x+1)(x-2)<BR>\n" );
document.write( "(x^2-4x+8)(x^2-2x+1x-2)<BR>\n" );
document.write( "(x^2-4x+8)(x^2-x-2)<BR>\n" );
document.write( "x^2(x^2-x-2)-4x(x^2-x-2)+8(x^2-x-2)<BR>\n" );
document.write( "x^4-x^3-2x^2-4x^3+4x^2+8x+8x^2-8x-16<BR>\n" );
document.write( "x^4-5x^3+10x^2-16\r<BR>\n" );
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document.write( "Therefore, the polynomial<BR>\n" );
document.write( "P(x) = x^4-5x^3+10x^2-16<BR>\n" );
document.write( "has the roots of x = 2-2i, x = 2+2i, x = -1, x = 2<BR>\n" );
document.write( "This can be confirmed using a tool like WolframAlpha.\r<BR>\n" );
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document.write( "Answer: <font color=red>P(x) = x^4-5x^3+10x^2-16</font><BR>\n" );
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