What is the polynomial function of lowest degree with lead coefficient 1 and roots i, –2, and 2?

1 Answer
Jun 16, 2018

#p(x)=x^4-3x^2-4#

Explanation:

#"given the root of a polynomial, say "x=a#

#"then "(x-a)" is a factor of the polynomial"#

#"complex roots occur in conjugate pairs"#

#x=i" is a root then so is "x=-i#

#"the polynomial is the product of it's factors"#

#p(x)=(x+2)(x-2)(x+i)(x-i)#

#"expanding the factors in pairs"#

#color(white)(p(x))=(x^2-4)(x^2-i^2)#

#[i^2=(sqrt(-1))^2=-1]#

#=(x^2-4)(x^2+1)#

#=x^4+x^2-4x^2-4#

#p(x)=x^4-3x^2-4#