A cubic polynomial with rational coefficients has the roots 6 +square root sixand 2 over 3. Find one additional root.
(1 point)
Responses
–6 –square root six
–6 – Image with alt text: square root six
6 +square root six
6 + Image with alt text: square root six
–6 +square root six
–6 + Image with alt text: square root six
6 –square root six
6 – Image with alt text: square root six
To find the additional root, we know that a cubic polynomial has three roots. Since two of the roots are given as 6 + √6 and 2/3, we can use the fact that the coefficients of the polynomial are rational to determine the third root.
Let's call the third root x. According to the rational root theorem, x must be a rational number of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. In this case, since we only have the roots and not the polynomial itself, we don't have the constant term or the leading coefficient.
However, since the cubic polynomial has rational coefficients, its non-rational roots must come in conjugate pairs. This means that if 6 + √6 is a root, then 6 - √6 must also be a root. Similarly, if 2/3 is a root, then -2/3 must also be a root.
Therefore, the additional root must be -6 - √6. So the correct answer is:
-6 - √6