anonymous
  • anonymous
MEDALS AWARDED!!!!! Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form. 4, -8, and 2 + 3i
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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DebbieG
  • DebbieG
Each root c corresponds to a factor, (x-c) Also, for any complex root a + bi, the complex conjugate a - bi must also be a root. So that means the 3 roots you're given REALLY tell you FOUR roots.... so line up those 4 corresponding factors: (x-4)(x+8)[x-(2 + 3i)][x-(2 - 3i)] And multiply it out.
anonymous
  • anonymous
yeah, you can multiply out, but there are two easier ways to find the quadratic with zeros \(2+3i\) and \(2-3i\)
anonymous
  • anonymous
one way is to work backwards set \[x=2+3i\] subtract \(2\) and get \(x-2=3i\) square to get \[(x-2)^2=-9\] or ' \[x^2-4x+4=-9\] then add 9 to get \[x^2-4x+13\] as your quadratic

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anonymous
  • anonymous
the real real easy way, although it requires memorization, is to know that if the quadratic has zeros \(a+bi\) and \(a-bi\) then it is \[x^2-2ab+(a^2+b^2)\] but you have to memorize that it is not hard though and would save you lots of time on a quiz or test
anonymous
  • anonymous
oops that was wrong it is \[x^2-2ax+(a^2+b^2)\]
anonymous
  • anonymous
so in your example since \(a+bi=2+3i\) you get \[x^2-2\times 2x+(2^2+3^2)=x^2-4x+13\]
DebbieG
  • DebbieG
Nice, that's slick! Great tip. I know that you can make the complex factors easier to multiply by rewriting as a product of sum and difference: [x-(2 + 3i)][x-(2 - 3i)]= [(x-2) - 3i)][(x-2) + 3i)] Then you get: \((x-2)^2 - (3i)^2=x^2-4x+4-(-9)=x^2-4+13\) which is kind of a similar idea, but I think I might like the "backwards" way that @satellite73 showed. :)
DebbieG
  • DebbieG
.... even better, I mean. lol
anonymous
  • anonymous
pretty much the exact same computation right?
anonymous
  • anonymous
i.e. you end up squaring \(x-2\) and then adding \(9\)
anonymous
  • anonymous
thank you i got it!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

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