Question

If 2 + √3 is a polynomial root, name another root of the polynomial, and explain how you know it must also be a root...

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find another root of the polynomial, we can use the fact that complex roots occur in conjugate pairs.

Let's assume that the given polynomial is a quadratic polynomial. In that case, if 2 + √3 is one root, then its conjugate 2 - √3 must also be a root. This is because the coefficients of the polynomial are real numbers, and if one root is a complex number, then its conjugate must also be a root to ensure that the polynomial has real coefficients.

To confirm that 2 - √3 is a root, we can substitute it into the polynomial equation. Let's say the quadratic polynomial is represented by ax^2 + bx + c.

Substituting 2 - √3 into the polynomial equation will give us:
a(2 - √3)^2 + b(2 - √3) + c

Expanding this expression, we get:
a(4 - 4√3 + 3) + b(2 - √3) + c
= 7a - 4a√3 + b(2 - √3) + c

Since 2 + √3 is assumed to be a root, we know that when substituted into this equation, the result should be zero. Therefore:
7a - 4a√3 + b(2 - √3) + c = 0

Now, substituting 2 - √3 into the equation:
7a - 4a√3 + b(2 - √3) + c
= 7a - 4a√3 + 2b - b√3 + c

Since this expression should also equal zero for 2 - √3 to be a root, we can conclude that:
7a - 4a√3 + 2b - b√3 + c = 0

Thus, 2 - √3 is another root of the polynomial.