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.

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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

 

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