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.
If 2 + √3 is a polynomial root, it means that when this value is substituted into the polynomial, the polynomial will evaluate to zero. Let's assume the polynomial is denoted as P(x).
To find another root, we can apply the concept of conjugate roots. For any polynomial with real coefficients, if a + b√c is a root of the polynomial, then the conjugate a - b√c will also be a root.
Therefore, if 2 + √3 is a root, its conjugate 2 - √3 must also be a root.
This can be explained using the properties of conjugate roots. When we substitute 2 + √3 into the polynomial P(x), it evaluates to zero. Thus, we have:
P(2 + √3) = 0
Now, if we substitute the conjugate, 2 - √3, into the polynomial, it should also evaluate to zero:
P(2 - √3) = 0
This connection between a root and its conjugate arises due to the fact that if we have a polynomial with real coefficients, irrational roots will always come in conjugate pairs.