find the remaining zeros of f.
degree 4 zeros i, -18+i
enter the remaining zeros of f
please help me out!
complex roots always come in conjugate pairs,
so the roots are ±i and -18+i and -18-i
so you would have
(x+i)(x-i)(x+ 18-i)(x+18+i)=0
(x^2 + 1)(x^2 + 18x + xi + 18x + 324 + 18i - xi - 18i - i^2) = 0
(x^2 + 1)(x^2 + 36x +324 + 1) = 0
(x^2 + 1)(x^2 + 36x + 325) = 0
expand again, and simplify
Well, first of all, let's appreciate the fact that the zeros of a polynomial are very loyal. They never leave without saying goodbye! Now, in this case, we already know that two zeros are i and -18+i. That's quite a party for the imaginary numbers! Now, for the remaining zeros, let's call them x and y. I mean, they must be feeling a bit left out, right? So, x and y, wherever you are, please come forward and show yourselves!
To find the remaining zeros of a polynomial function with a degree of 4, we can use the fact that complex zeros occur in conjugate pairs. Since the zeros i and -18+i are given, we know that they are conjugates of each other.
Conjugate pairs mean that if one zero is a + bi, then the other zero is a - bi.
Therefore, if -18 + i is a zero, then its conjugate, -18 - i, must also be a zero.
Hence, the remaining zeros of f are: -18 - i.
To find the remaining zeros of the function f, we need to first determine the polynomial that represents the function.
You have provided two zeros: i and -18+i. Since complex roots occur in conjugate pairs, we know that the conjugate of -18+i is -18-i.
Given these zeros, we can write the polynomial as follows:
f(x) = (x - i)(x + 18 - i)(x + 18 + i)(x - (-18 + i))
To find the remaining zeros, we need to solve the equation f(x) = 0.
Setting f(x) equal to zero, we have:
(x - i)(x + 18 - i)(x + 18 + i)(x + 18 - i) = 0
To find the remaining zeros, we can set each factor equal to zero and solve for x:
x - i = 0 --> x = i
x + 18 - i = 0 --> x = -18 + i
x + 18 + i = 0 --> x = -18 - i
x - (-18 + i) = 0 --> x = 18 - i
So, the remaining zeros of f are: i, -18 + i, -18 - i, and 18 - i.
Therefore, the complete set of zeros for f is: i, -18 + i, -18 - i, and 18 - i.