form the polynomial given,degree 4 and zeros: i, 1+2i
I would use those roots, and the two conjugate roots, then muliply it out.
To form a polynomial with degree 4 and the given zeros, we need to consider both the real and imaginary parts of the zeros.
Let's start with the imaginary zero i. Complex zeros always come in conjugate pairs, so the conjugate of i is -i. Hence, we have two zeros: i and -i.
The factors for these two zeros are (x - i) and (x + i) respectively.
Now, let's consider the zero 1+2i. Since this is not a conjugate of an existing zero, it implies that the polynomial is not restricted to real coefficients. Therefore, we consider the zero 1-2i as well, which is the conjugate of 1+2i.
The factors for these two zeros are (x - (1+2i)) and (x - (1-2i)) respectively.
Simplifying each factor, we get:
(x - i)(x + i) = x^2 + 1
(x - (1+2i))(x - (1-2i)) = (x - 1 - 2i)(x - 1 + 2i) = (x - 1)^2 - (2i)^2 = x^2 - 2x + 1 - 4 = x^2 - 2x - 3
Now, we multiply these factors together to form the polynomial:
(x^2 + 1)(x^2 - 2x - 3) = x^4 - 2x^3 - 3x^2 + x^2 - 2x - 3 = x^4 - 2x^3 - 2x^2 - 2x - 3
Therefore, the polynomial with degree 4 and zeros i and 1+2i is:
f(x) = x^4 - 2x^3 - 2x^2 - 2x - 3.
To form a polynomial given its degree and zeros, you need to use the concept of complex conjugate pairs, as complex roots always occur in conjugate pairs. The given zeros are "i" and "1+2i".
Since "i" is a zero, its conjugate, "-i", is also a zero. Similarly, since "1+2i" is a zero, its conjugate, "1-2i", is also a zero.
To form the polynomial, you can multiply the factors corresponding to each zero. The factors are obtained by subtracting each zero from the variable "x". For "i" as a zero, the factor is "(x - i)". For "-i", it is "(x + i)". For "1+2i", it is "(x - (1+2i)) = (x - 1 - 2i)". And for "1-2i", it is "(x - (1-2i)) = (x - 1 + 2i)".
Now, to form the polynomial, you multiply all the factors together:
(x - i)(x + i)(x - 1 - 2i)(x - 1 + 2i)
Expanding this expression will result in a polynomial with a degree of 4.
(x^2 - i^2)(x^2 - 1 + 2ix - 1 - 2ix + 4)
Simplifying further by using the identity "i^2 = -1":
(x^2 + 1)(x^2 - 2x + 3)
Finally, multiplying the remaining factors:
x^4 - 2x^3 + 3x^2 + x^2 - 2x + 3
Combining like terms:
x^4 - 2x^3 + 4x^2 - 2x + 3
Therefore, the polynomial of degree 4 with zeros "i" and "1+2i" is:
f(x) = x^4 - 2x^3 + 4x^2 - 2x + 3