how do you write simples polynomial function with the given zero
1-i and 2
(x-1+i)(x-2)=0
multiply it out, and gather terms.
However, if you want a polynomial with real coefficients, you need another zero: the conjugate of 1-i = 1+i
SO, your polynomial is then
(x-(1-i))(x-(1+i))(x-2)
((x-1)+i)((x-1)-i)(x-2)
((x-1)^2+1)(x-2)
(x^2-2x+2)(x-2)
...
To write a polynomial function with the given zeros, we can use the concept of complex conjugate pairs.
1. Given the zero 1-i: Since complex zeros always occur in conjugate pairs, the conjugate of 1-i is 1+i. Therefore, we have two zeros: 1-i and 1+i.
2. Given the zero 2.
Now, we can write the polynomial function by multiplying out the factors with the given zeros:
(x - (1-i))(x - (1+i))(x - 2)
First, let's simplify (x - (1-i))(x - (1+i)):
(x - 1 + i)(x - 1 - i) = (x^2 - x + xi - x + 1 - i - xi + i - i^2)
= (x^2 - 2x + 2)
Now we can multiply the simplified expression with the third factor:
(x^2 - 2x + 2)(x - 2)
Expanding and combining like terms, we get:
x^3 - 2x^2 + 2x - 2x^2 + 4x - 4 + 2x - 4
= x^3 - 4x^2 + 8x - 8
Therefore, the polynomial function with the given zeros 1-i, 1+i, and 2 is f(x) = x^3 - 4x^2 + 8x - 8.