If a polynomial function with rational coefficients has the zeros -5 and i, what are the additional zeros?

1 Answer
Nov 15, 2016

Answer:

Additional zero is #-i# and polynomial function is #x^3+5x^2+x+5#

Explanation:

A polynomial function whose zeros are #alpha#, #beta#, #gamma# and #delta# and multiplicities are #p#, #q#, #r# and #s# respectively is

#(x-alpha)^p(x-beta)^q(x-gamma)^r(x-delta)^s#

It is apparent that the highest degree of such a polynomial would be #p+q+r+s#.

In the given case as two zeros are #-5# and #i# and coefficients are rational, assuming multiplicity of each to be just #1#,

we will have to have additionally a zero #-i# (being complex conjugate of #i#) and hence polynomial is

#(x-(-5))(x-i)(x-(-i))#

= #(x+5)(x-i)(x+i)#

= #(x+5)(x^2+1)#

= #x^3+5x^2+x+5#