What is the polynomial function of lowest degree with lead coefficient 1 and roots i, –2, and 2?
1 Answer
Jun 16, 2018
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#