Given that the polynomial function (below) has the given zero, find the other zeros? #f(x) = x^4 - 5x^3 + 7x^2 - 5x + 6; -i#

I know the answers are #i, 2, 3# but I don't know how to find them.

1 Answer
Douglas K.
Nov 16, 2017

The general factored form of a quartic equation is:

#y = k(x-r_1)(x-r_2)(x-r_3)(x-r_4)" [1]"#

Where #r_1,r_2,r_3, and r_4# are the roots of the equation.

Explanation:

Given: #f(x) = x^4 - 5x^3 + 7x^2 - 5x + 6; -i#

We can substitute #-i# for #r_1# into equation [1]:

#y = k(x+i)(x-r_2)(x-r_3)(x-r_4)" [1.1]"#

Because all of the coefficients are real, we know that imaginary roots must exist in conjugate pairs, therefore, #i# must, also, be a root. Substitute #i# for #r_2# in equation [1.1]:

#y = k(x+i)(x-i)(x-r_3)(x-r_4)" [1.2]"#

Because the leading coefficient of the given equation is 1, we know that #k = 1#, therefore, remove k from equation [1.2]:

#y = (x+i)(x-i)(x-r_3)(x-r_4)" [1.3]"#

We know that #(x-i)(x+i) = x^2+1#, therefore, the original polynomial must be evenly divisible by #x^2+1#:

#(x^4 - 5x^3 + 7x^2 - 5x + 6)/(x^2+1) = x^2-5x+6 larr# the factors of this quadratic must contain the remaining roots:

#x^2-5x+6 = (x-2)(x-3)#

#y = (x+i)(x-i)(x-2)(x-3)" [1.4]"#

The roots are: #-i, i, 2, and 3#

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