Question 189408: Show that 7 + √2 is an algebraic number
Answer by solver91311(24713)   (Show Source):
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An algebraic number is a complex number that is a root of a non-zero polynomial in one variable with rational (or equivalently, integer) coefficients.
So, if you can show that  is the root of a non-zero polynomial with rational coefficients, then must be an algebraic number.
Irrational roots of quadratic polynomials always come in conjugate pairs, that is if a is rational, b is irrational, and a + b is a root of  then a - b must also be a root of .
So, if  is a root, then  must be the other root, and the factors of the desired polynomial, if it exists, must be:
)(x - (7 - sqrt{2})))
Applying FOIL:
x - (7 + sqrt{2})x + (49 - 2))
which is a non-zero polynomial in one variable with integer, and therefore rational, coefficients, and therefore is an algebraic number.
John
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