Determine the polynomial function The zeroes of polynomial are 1-√3,1 + √3 - 2

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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Since the polynomial has 3 zeroes, that means that the polynomial has 3 roots, then the polynomial is of 3rd degree.

f(x) = ax^3 + bx^2 + cx + d

According to the rule, a polynomial could be written as product of linear factors, also.

Since the roots of polynomial are:

x1 = 1 - sqrt3

x2 = 1 + sqrt3

x3 = -2

f(x) = (x - 1 + sqrt3)(x - 1 - sqrt3)(x + 2)

f(x) = [(x-1)^2 - 3](x+2)

f(x) = (x^2 - 2x + 2 - 3)(x+2)

f(x) = (x^2 - 2x - 1)(x+2)

We'll remove the brackets and we'll get:

f(x) = x^3 + 2x^2 - 2x^2 - 4x - x - 2

We'll eliminate and combine like terms:

f(x) = x^3 - 5x - 2

The function that has the 3 given roots is:

f(x) = x^3 - 5x - 2

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