Write a polynomial function given the zeroes:
11i and -11i
p(x) = (x-11i)(x+11i) = x^2 - (11i)^2 = x^2+121
When a polynomial function has complex roots, the conjugate property of complex numbers comes into play. If 11i is a root of the polynomial, then its conjugate -11i is also a root.
To write the polynomial function, we can use the fact that the complex roots occur in conjugate pairs. The quadratic factors can be written as (x - 11i)(x + 11i) since (11i) and (-11i) are conjugates, and when multiplied together, they produce a quadratic term with no imaginary part.
Expanding the quadratic factors, we get:
(x - 11i)(x + 11i) = x^2 - (11i)^2
Remember that i^2 = -1, so we can simplify:
x^2 - (11i)^2 = x^2 - 121
Therefore, the polynomial function with the given zeroes is:
f(x) = (x - 11i)(x + 11i)
= x^2 - 121
To write a polynomial function given the zeros, we need to take the complex conjugates of the zeros and use them as factors in the polynomial. The complex conjugate of `11i` is `-11i`, and vice versa.
So, the factors for the polynomial are `(x - 11i)` and `(x + 11i)`.
To write the polynomial, multiply these factors together:
`(x - 11i)(x + 11i)`
Now, let's expand this expression:
`(x - 11i)(x + 11i) = x * x + x * 11i - x * 11i - 11i * 11i`
Remember that `(a + b)(a - b) = a^2 - b^2`. Applying this pattern to our expression, we can simplify it further:
`= x^2 + 11ix - 11ix - (11i)^2`
Since `i^2` is equal to `-1`, we can simplify further:
`= x^2 - (11i * 11i)`
`= x^2 - 121*(-1)`
`= x^2 + 121`
Hence, the polynomial function with zeros `11i` and `-11i` is `f(x) = x^2 + 121`.