If you know that 3+√11 is a root of a polynomial function, then the name given to 3−√11 , another root of the same function , is a __ conjugate. ?

1 Answer
George C.
Dec 17, 2017

radical conjugate

Explanation:

A conjugate is an object which when combined with the original object makes some kind of whole.

Note that the difference of squares identity tells us that:

a2−b2=(a−b)(a+b)

So if a and b are terms consisting of square roots (possibly including i which is a square root of −1), then we can simplify (a+b) by multiplying by the conjugate (a−b) (or vice versa).

In the case of square roots, this is a radical conjugate.

For example, we find:

(x−(3−√11))(x−(3+√11))=((x−3)−√11)((x−3)+√11)

(x−(3−√11))(x−(3+√11))=(x−3)2−(√11)2

(x−(3−√11))(x−(3+√11))=x2−6x+9−11

(x−(3−√11))(x−(3+√11))=x2−6x−2

Note that the resulting quadratic polynomial has only rational coefficients - we have successfully eliminated the irrational √11 by using the radical conjugate.

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