Find the quadratic polynomial whose one zero is 2+sqrt(3)

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embizze | High School Teacher | (Level 2) Educator Emeritus

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Find a quadratic polynomial with rational coefficients with `(2+sqrt(3))` as a zero:

If `2+sqrt(3)` is a zero, so is the conjugate `2-sqrt(3)` .

Also, if a is a zero, then (x-a) is a factor, thus the factors of the quadratic are:


Multiplying we get:


Adding like terms:



The monic quadratic polynomial with rational coefficients is:



** Checking we can use the quadratic formula to find the roots(zeros):





and we see that `2+sqrt(3)` is a zero.

*** Note that this is not the only such quadratic polynomial, even if we restrict the coefficients to be rational. If the leading coefficient need not be one (need not be a monic polynomial), then `2x^2-8x+2` works also; there are an infinite number of such polynomials.


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