Elenathehomeschooler
  • Elenathehomeschooler
Someone help! Write a function with the given characteristics. A polynomial with rational coefficients having roots 4 and 3-i
Mathematics
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SOLVED
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schrodinger
  • schrodinger
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Elenathehomeschooler
  • Elenathehomeschooler
@jim_thompson5910 can you help?
jim_thompson5910
  • jim_thompson5910
Hopefully you agree that if 3-i is one root, another root is 3+i, right?
Elenathehomeschooler
  • Elenathehomeschooler
yes i agree wch would make our roots 4 , 3-i , and 3+i right?

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Elenathehomeschooler
  • Elenathehomeschooler
@jim_thompson5910
jim_thompson5910
  • jim_thompson5910
if 4 is a root, then do you agree that x-4 is a factor?
Elenathehomeschooler
  • Elenathehomeschooler
yes
jim_thompson5910
  • jim_thompson5910
if 3+i and 3-i is a root, then what must the factor be? any idea?
Elenathehomeschooler
  • Elenathehomeschooler
3?
jim_thompson5910
  • jim_thompson5910
If 3+i is a root, then... x = 3+i x - 3 = i (x - 3)^2 = i^2 (x - 3)^2 = -1 (x - 3)^2 + 1 = 0 Hopefully you see how if 3+i is a root, then (x - 3)^2 + 1 is factor
Elenathehomeschooler
  • Elenathehomeschooler
ok
jim_thompson5910
  • jim_thompson5910
agreed? or no?
Elenathehomeschooler
  • Elenathehomeschooler
yes i agree
jim_thompson5910
  • jim_thompson5910
so we have these two factors (x-4) and ((x - 3)^2 + 1)
jim_thompson5910
  • jim_thompson5910
(x - 3)^2 + 1 is the same as x^2-6x+9+1 = x^2-6x+10
Elenathehomeschooler
  • Elenathehomeschooler
would we end up FOIL them?
jim_thompson5910
  • jim_thompson5910
so you need to multiply out (x-4) and (x^2-6x+10)
Elenathehomeschooler
  • Elenathehomeschooler
would it be x^3-10x^2+34x-40
jim_thompson5910
  • jim_thompson5910
that is the correct answer
Elenathehomeschooler
  • Elenathehomeschooler
so that would be the final answer?
jim_thompson5910
  • jim_thompson5910
yes

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