How to prove that a value is or isn't the root of a polynomial?Example of how to show that a value is / isn't the root of a polynomial.
We'll take the value x = sqrt 3 - i and f(x) = x^4-4x^2+16 and we'll prove that f(x)=0.
If f(x)=0, then x = sqrt3 - i is the root of the polynomial f(x).
We'll substitute x by a and we'll verify if f(x) = 0
f(x) = x^4 – 4x^2 + 16
f(x) = x^2(x^2 - 4) + 16
We'll re-write the difference of squares x^2 - 4 = (x-2)(x+2)
x = sqrt3 - i
We'll square raise both sides:
x^2 = (sqrt3 - i)^2
We'll expand the square:
x^2 = 3 - 2isqrt3 + i^2, where i^2 = -1
x^2 = 2 - 2isqrt3
f(sqrt3 - i) = (2 - 2isqrt3)(2 - 2isqrt3 - 4) + 16
We'll combine like terms inside brackets:
f(sqrt3 - i) = (2 - 2isqrt3)(-2 - 2isqrt3) + 16
f(sqrt3 - i) = -(2 - 2isqrt3)(2 + 2isqrt3) + 16
We'll write the product as a difference of squares:
f(sqrt3 - i) = -(2^2 - 4*3*i^2) + 16