If 2 + square root 3 is a polynomial root, name another root of the polynomial, and explain how you know it must also be a root.
To find another root of the polynomial, we will use the fact that if a polynomial has a root, say r, then the conjugate of r, denoted by r', must also be a root.
Given that 2 + √3 is a root of the polynomial, this implies that its conjugate, 2 - √3, must also be a root.
To prove this, consider the polynomial equation P(x) = 0, where P(x) is the polynomial in question.
Let r = 2 + √3 be a root of P(x), then we have:
P(r) = P(2 + √3) = 0
Now, let's consider the conjugate, r' = 2 - √3. The conjugate of a number is formed by changing the sign of the square root part (in this case, √3).
If we substitute r' into the polynomial, we get:
P(r') = P(2 - √3)
Now, let's manipulate this expression:
P(2 - √3) = P(2 + √3 - 2√3)
= P(2 + √3 - 2√3 + √3)
= P(2 - √3 + √3)
= P(2)
Since P(r) = 0 (as r is a root), we can rewrite P(2) = 0.
Thus, we have P(r') = P(2) = 0, which means that r' = 2 - √3 is also a root of the polynomial.
Therefore, 2 - √3 is another root of the polynomial, and we can conclude that if 2 + √3 is a root, then 2 - √3 must also be a root.