document.write( "Question 706140: hi please help me solve this question
\n" ); document.write( "7. (a) Solve the equation z^4-2z^3+5z^2-6z+6=0 given it has a root of the form z=ia where a is a real number.
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\n" ); document.write( " (b) Evaluate the sum i+i^2+i^3+i^4+....+i^174 \n" ); document.write( "
Algebra.Com's Answer #435005 by KMST(5328)\"\" \"About 
You can put this solution on YOUR website!
(a) \"z%5E4-2z%5E3%2B5z%5E2-6z%2B6=0+\"
\n" ); document.write( "\"i%5E2=-1\" , \"i%5E3=-i\" and \"i%5E4=1\"
\n" ); document.write( "If \"z=ia\" then
\n" ); document.write( "\"z%5E2=%28ia%29%5E2=i%5E2a%5E2=%28-1%29%28a%5E2%29=-a%5E2\" ,
\n" ); document.write( "\"z%5E3=%28ia%29%5E3=i%5E3a%5E3=-ia%5E3\" and
\n" ); document.write( "\"z%5E4=%28ia%29%5E4=i%5E4a%5E4=1%2Aa%5E4=a%5E4\"
\n" ); document.write( "If \"z=ia\" is a root, then substituting into \"z%5E4-2z%5E3%2B5z%5E2-6z%2B6=0\" we get
\n" ); document.write( "\"a%5E4%2Bi%282a%5E3%29-5a%5E2%2Bi%28-6a%29%2B6=0\" --> \"%28a%5E4-5a%5E2%2B6%29%2Bi%282a%5E3-6a%29=0\" --> \"%28a%5E4-5a%5E2%2B6%29%2Bi%282a%29%28a%5E2-3%29=0\" --> \"%28%28a%5E2-3%29%28a%5E2-2%29%29%2Bi%282a%29%28a%5E2-3%29=0\"
\n" ); document.write( "That means that \"%28a%5E2-3%29%28a%5E2-2%29=0\" and \"%282a%29%28a%5E2-3%29=0\"
\n" ); document.write( "at the same time (for the same real \"a\" value).
\n" ); document.write( "\"highlight%28a=sqrt%283%29%29\" and \"highlight%28a=-sqrt%283%29%29\" are the only solutions.
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\n" ); document.write( "EXTRA:
\n" ); document.write( "The imaginary roots are \"z=-i%2Asqrt%283%29\" and \"z=i%2Asqrt%283%29\".
\n" ); document.write( "so a factor in the factoring of \"z%5E4-2z%5E3%2B5z%5E2-6z%2B6\" should be
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\n" ); document.write( "Dividing we find that
\n" ); document.write( "\"z%5E4-2z%5E3%2B5z%5E2-6z%2B6=%28z%5E2%2B3%29%28z%5E2-2z%2B2%29\"
\n" ); document.write( "So the other two complex solutions to \"z%5E4-2z%5E3%2B5z%5E2-6z%2B6=0\"
\n" ); document.write( "are the solutions to \"z%5E2-2z%2B2=0\"
\n" ); document.write( "which happen to be \"z=1+%2B-+i\"
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\n" ); document.write( "(b)
\n" ); document.write( "ONE WAY TO LOOK AT IT:
\n" ); document.write( "\"i%5E0=1\" , \"i%5E1=i\" , \"i%5E2=-1\" , \"i%5E3=-i\" , \"i%5E4=1\" , \"i%5E5=i\" , \"i%5E6=-1\" and so on.
\n" ); document.write( "In general, for any integer \"n\"
\n" ); document.write( "\"i%5E%284n%29=1\" , \"i%5E%284n%2B1%29=i\" , \"i%5E%284n%2B2%29=-1\" , \"i%5E%284n%2B3%29=-i\"
\n" ); document.write( "and \"i%5E%284n%29%2Bi%5E%284n%2B1%29%2Bi%5E%284n%2B2%29%2Bi%5E%284n%2B3%29=1%2Bi-1-i=0\"
\n" ); document.write( "So adding up the terms of the sum in groups of 4 we get
\n" ); document.write( "\"i%2Bi%5E2%2Bi%5E3%2Bi%5E4=1%2Bi-1-i=0\" , \"i%5E5%2Bi%5E6%2Bi%5E7%2Bi%5E8=1%2Bi-1-i=0\" and so on.
\n" ); document.write( "Until when?
\n" ); document.write( "Dividing 174 by 4 we get a remainder of 2:
\n" ); document.write( "\"174=172%2B2=4%2A24%2B2\" so the last group of 4 would end with \"i%2A172=i%5E%284%2A24%29=1\"
\n" ); document.write( "After that we have \"i%5E173=i%5E%284%2A24%2B1%29=i\" and \"i%2A174=i%5E%284%2A24%2B2%29=-1\" , so
\n" ); document.write( "\"i%2Bi%5E2%2Bi%5E3%2Bi%5E4\"+....+\"i%5E174+=+%28i%2Bi%5E2%2Bi%5E3%2Bi%5E4%29\"+....+\"%28i%5E169%2Bi%5E170%2Bi%5E171%2Bi%5E172%29%2Bi%5E173%2Bi%5E174+=+0%2B0%2B0\"+...+\"0%2Bi%5E173%2Bi%5E174=highlight%28i-1%29\"
\n" ); document.write( "or if you prefer \"highlight%28-1%2Bi%29\"
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\n" ); document.write( "ANOTHER WAY:
\n" ); document.write( "i+i^2+i^3+i^4+....+i^174 is the sum of a geometric sequence,(or geometric progression, depending on where you are studying math).
\n" ); document.write( "From what we may know about geometric sequences and series,
\n" ); document.write( "i+i^2+i^3+i^4+....+i^174 = \"%28i%5E175-i%29%2F%28i-1%29\"
\n" ); document.write( "Since \"i%5E175=i%5E%284%2A24%2B3%29=-i\" ,
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