write a polynomial function in standard form with the given zeros. 1-2i,2
no comma there actually
zero when
x = 1-2i
x = 1+2i
x = 2
so I get
(x-2)(x-1+2i)(x-1-2i)
(x-2) (x^2-2x+5)
complex numbers, like radicals, always come in their conjugate form
so if -2i is a zero so is +2i
so f(x) = (x-1)(x-2)(x - 2i)(x + 2i)
the last two factors multiply for x^4 + 4
f(x) = (x-1)(x-2)(x^4 + 4)
expand if needed, I like it the way it is.
Just have to get stronger glasses.
To find a polynomial function in standard form with the given zeros, we use the fact that complex zeros occur in conjugate pairs. Given the zeros 1-2i and 2, we know that the conjugate of 1-2i is 1+2i.
To form a polynomial with these zeros, we need to include factors corresponding to each zero. The factor for 1-2i and its conjugate would be (x - (1-2i))(x - (1+2i)), which simplifies to (x - 1 + 2i)(x - 1 - 2i).
Next, we need to account for the zero 2. The factor for this zero is simply (x - 2).
Finally, we can multiply all the factors together to obtain the polynomial function in standard form.
P(x) = (x - 1 + 2i)(x - 1 - 2i)(x - 2)
Expanding this expression further, we get:
P(x) = (x^2 - 2x + 5)(x - 2)
Multiplying again, we obtain:
P(x) = x^3 - 2x^2 + 5x - 2x^2 + 4x - 10
Combining like terms, the polynomial function in standard form is:
P(x) = x^3 - 4x^2 + 9x - 10