document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204777>Question 1204777</A>: <I><SPAN STYLE=\"word-break: break-all;\"> form a polynomial f(x) with real coefficients having the given degree and zeros degree :4, zeros: 5+2i, -1 multiplicity 2 </SPAN>\n" );
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document.write( "Answer: <font color=red>f(x) = x^4  - 8x^3 + 10x^2 + 48x + 29</font>\r<BR>\n" );
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document.write( "Explanation\r<BR>\n" );
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document.write( "The polynomial has real number coefficients, so any complex root of the form a+bi has a conjugate a-bi\r<BR>\n" );
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document.write( "The root 5+2i pairs with 5-2i<BR>\n" );
document.write( "x = 5+2i<BR>\n" );
document.write( "x-5 = 2i<BR>\n" );
document.write( "(x-5)^2 = (2i)^2<BR>\n" );
document.write( "(x-5)^2 = 4i^2<BR>\n" );
document.write( "(x-5)^2 = 4(-1)<BR>\n" );
document.write( "(x-5)^2 = -4<BR>\n" );
document.write( "(x-5)^2+4 = 0<BR>\n" );
document.write( "x^2-10x+25+4 = 0<BR>\n" );
document.write( "x^2-10x+29 = 0<BR>\n" );
document.write( "You'll arrive at this same equation if you started with x = 5-2i\r<BR>\n" );
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document.write( "Therefore x^2-10x+29 = 0 has the complex roots x = 5+2i and x = 5-2i<BR>\n" );
document.write( "The quadratic formula can be used to confirm this. <BR>\n" );
document.write( "Online CAS (computer algebra system) tools such as WolframAlpha or GeoGebra can also be used to confirm this claim.\r<BR>\n" );
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document.write( "So far we have shown that (x^2-10x+29) is a factor\r<BR>\n" );
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document.write( "If x = -1 is a root then (x+1) is a factor<BR>\n" );
document.write( "This root is of multiplicity 2. It's a double root. So (x+1)^2 is a factor.\r<BR>\n" );
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document.write( "Let's expand out the following<BR>\n" );
document.write( "(x+1)^2*(x^2-10x+29)<BR>\n" );
document.write( "(x^2+2x+1)*(x^2-10x+29)<BR>\n" );
document.write( "x^2(x^2-10x+29) + 2x(x^2-10x+29) + 1(x^2-10x+29)<BR>\n" );
document.write( "(x^4-10x^3+29x^2) + (2x^3-20x^2+58x) + (x^2-10x+29)<BR>\n" );
document.write( "x^4 + (-10x^3+2x^3) + (29x^2-20x^2+x^2) + (58x-10x) + 29<BR>\n" );
document.write( "<font color=red>x^4  - 8x^3 + 10x^2 + 48x + 29</font>\r<BR>\n" );
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document.write( "Using a CAS is one way to confirm the answer above is correct.\r<BR>\n" );
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document.write( "Technically we could scale this polynomial up or down, but I'll leave the leading coefficient as 1.<BR>\n" );
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