use given info about a polynomial whose coefficients are real numbers to find the remaining zeros.
Degree: 6 so I know there's at least 6
zeros:-5-isqrt7, 13 + 2ni, -5 - 3i
(where n is a real number)
any ideas??????
The idea is to remember that you're dealing with a polynomial with real coefficients. That means that any complex roots occur in conjugate pairs.
So, that means that the six roots are -5-isqrt7 -5+isqrt7
13+2ni 13-2ni
-5-3i -5+3i
Now you just multiply all the factors together to get the polynomial.
To find the remaining zeros of a polynomial, we can use the fact that complex conjugates always occur in pairs.
Given the zeros:
-5 - i√7 (complex number)
13 + 2ni (complex number)
-5 - 3i (complex number)
We can see that there are two complex zeros: -5 - i√7 and -5 - 3i. By the conjugate pair theorem, their conjugates will also be zeros of the polynomial.
Complex conjugates occur when the imaginary parts are the same, but with opposite signs.
For the zero -5 - i√7, its conjugate would be -5 + i√7.
For the zero -5 - 3i, its conjugate would be -5 + 3i.
So, the remaining zeros of the polynomial are:
-5 + i√7
-5 + 3i
Therefore, the zeros of the polynomial are:
-5 - i√7
-5 + i√7
-5 - 3i
-5 + 3i
Note: The fact that the coefficients of the polynomial are real numbers is not relevant to finding the remaining zeros using the complex conjugate theorem.