How do you find a third degree polynomial given roots #-7# and #i#?

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Answer 1 · 2018-07-17T20:10:07

#x^3+7x^2+x+7#

The imaginary or complex roots of a polynomial equation are always conjugate hence if #i# is a root then #-i# will also be another root.

Thus, the cubic polynomial #P(x)# has three roots #\alpha=-7, \ \beta=i, \ \ \gamma=-i# hence the cubic polynomial is given as

#(x-\alpha)(x-\beta)(x-\gamma)#

#=(x-(-7))(x-i)(x-(-i))#

#=(x+7)(x-i)(x+i)#

#=(x+7)(x^2-i^2)#

#=(x+7)(x^2+1)#

#=x^3+7x^2+x+7#