A polynomial function with rational coefficients has the following zeros. Find all additional zeros.
2, -2 + �ã10
To find the additional zeros of a polynomial function with rational coefficients, we can utilize the complex conjugate theorem. The complex conjugate theorem states that if a polynomial P(x) with real coefficients has a complex zero a + bi, then its conjugate a - bi is also a zero.
In this case, we have the zeros 2 and -2 + √10. The complex conjugate of -2 + √10 is -2 - √10. Therefore, we can conclude that the additional zero is -2 - √10.
Hence, the additional zeros are -2 - √10.
To find the additional zeros of a polynomial function, we need to use the fact that zeros come in conjugate pairs for polynomials with real coefficients.
Given the given zeros: 2 and -2 + √10
Since 2 is a real zero, it will have a conjugate of itself. Therefore, its conjugate is 2 - √10.
Similarly, for -2 + √10, its conjugate would be -2 - √10.
Now, we can use these new conjugate zeros to find the additional zeros.
Set each conjugate equal to zero and solve:
2 - √10 = 0
Solving for √10:
√10 = 2
So, one of the additional zeros is √10 = 2.
Next, solve for the other conjugate:
-2 - √10 = 0
Solving for √10:
√10 = -2
However, since we are looking for real zeros here, √10 cannot equal -2. Therefore, there are no additional real zeros. The only additional zero is √10 = 2.