Suppose that a polynomial function of degree 4 with rational coefficients has i and (-3 + square root of 3)as zeros find the other zeros
There must be four roots total and complex numbers have conjugates that are roots. One other root is therefore
-i .
If -3 + sqrt3, is a root, the another root is -3 - sqrt3, as a consequence of the +/-sqrt(b^2 - 4ac) in the quadratic equation.
The polynomial must be a multiple of
(x^2 +1)(x +3 -sqrt3)(x +3 +sqrt3) = 0
(x^2+1)[(x+3)^2 -3] = 0
(x^2+1)(x^2 +6x +6) = 0
x^4 +6x^3 +7x^2 +6 = 0
Well, the good news is that you already have two zeros of the polynomial: i and (-3 + √3). Now all we have to do is find the other two zeros!
But let's not make this too complicated. We can use the conjugate rule. Since (-3 + √3) is a zero, its conjugate (-3 - √3) must also be a zero. So, there's one more!
Now, since a polynomial of degree 4 can have at most 4 zeros, we have found all of them. So, the other zeros are i, (-3 + √3), and (-3 - √3).
I hope this helps, or at least brings a smile to your face!
To find the other zeros of the polynomial function, we need to consider the complex conjugates of the given zeros.
Given that the polynomial function has i as a zero, we know that its conjugate (-i) is also a zero.
Similarly, if (-3 + square root of 3) is a zero, then its conjugate (-3 - square root of 3) is also a zero.
Therefore, the other zeros of the polynomial function are -i and (-3 - square root of 3).
To find the other zeros of the polynomial function with degree 4, we need to use the fact that complex zeros occur in conjugate pairs.
First, let's consider the zero i. Since complex conjugates appear as pairs, the conjugate of i is -i. Therefore, the polynomial also has -i as a zero.
Next, we can focus on the zero (-3 + √3). Since it is not purely imaginary, its conjugate will be (-3 - √3). Thus, the polynomial also has (-3 - √3) as a zero.
Now that we have identified four zeros of the polynomial (i, -i, -3 + √3, and -3 - √3), we can employ a method called synthetic division or long division to divide the polynomial by (x - xi)(x + xi)(x - (-3 + √3))(x - (-3 - √3)), where xi represents the imaginary part i.
Performing the division process will give us a quotient polynomial of degree 0, which is a constant. This constant will then represent the last zero of the polynomial.
By using the algebraic process mentioned above, you can find the other zeros of the polynomial.