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3 votes
A cubic polynomial with rational coefficients has the roots 6 + *sqrt* 6 and 2/3 . Find one additional root.

-6-*sqrt*6
6+*sqrt*6
-6+*sqrt*6
6-*sqrt*6

2 Answers

2 votes

Answer:
6 - √(6)

Explanation:

The Conjugate Root Theorem states if P(x) is a polynomial with rational coefficients, then irrational roots of P(x) = 0 that have the form
a + √(b) occur in conjugate pairs. That is, if
a + √(b) is an irrational root with a and b rational, then
a - √(b) is also a root.

Since
6 + √(6) is a root, then
6 - √(6) is also a root.

User Youngmi
by
8.1k points
6 votes
The correct answer is 6 - sqrt(6)

We know this because when we solve with the quadratic equation, the numbers come out with a constant plus and minus the discriminant under a square root symbol. You can see that below in the quadratic equation.


\frac{-b +/- \sqrt{ b^(2) - 4ac } }{2a}

So seeing that one answer is 6 + sqrt(6), we can assume 6 is the discriminat and 6 is the constant. Thus giving us the new answer as well.
User Ben Rogmans
by
8.0k points

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