Given these zeros, write a polynomial function of least degree with integral coefficients.
2i, 2 + 2.236
2+2.236 = 4.236 = 4236/1000 =1059/250
complex roots come in pairs, so
(x-2i)(x+2i)(x - 1059/250)
(x^2+4) * 1/250 (250x - 1059)
1/250 (x^2+4)(250x-1059)
Why do I suspect a typo?
I suspect a typo but do
(x - 2i) (x - whatever that second zero is) = 0
To create a polynomial function with integral coefficients given the complex zeros, we need to also include the conjugates of those zeros.
Given the zeros:
1. 2i
2. 2 + 2.236
The conjugates of these zeros are:
1. -2i
2. 2 - 2.236
To find the polynomial function, we can multiply the factors of the form (x - zero) together.
For the zeros 2i and -2i, the factors are (x - 2i) and (x + 2i), respectively.
For the zeros 2 + 2.236 and 2 - 2.236, the factors are (x - (2 + 2.236)) and (x - (2 - 2.236)), respectively.
Simplifying each factor, we get:
1. (x - 2i)
2. (x + 2i)
3. (x - (2 + 2.236)) = (x - 4.236)
4. (x - (2 - 2.236)) = (x - (-0.236)) = (x + 0.236)
Finally, we multiply all the factors together:
(x - 2i)(x + 2i)(x - 4.236)(x + 0.236)
Expanding this expression, we get:
= (x^2 - (2i)^2)(x - 4.236)(x + 0.236)
= (x^2 + 4)(x - 4.236)(x + 0.236)
Therefore, the polynomial function of least degree with integral coefficients is:
f(x) = (x^2 + 4)(x - 4.236)(x + 0.236)
To write a polynomial function with integral coefficients, we need to consider the complex conjugates of the given zeros as well.
The complex conjugate of 2i is -2i.
Therefore, the zeros of the polynomial function are:
2i, -2i, 2 + 2.236, 2 - 2.236.
Since complex zeros always occur in conjugate pairs, we can combine the complex zeros as follows: 2i and -2i, 2 + 2.236 and 2 - 2.236.
The polynomial function will have factors that correspond to these zeros.
For the complex conjugate zeros 2i and -2i, the corresponding quadratic factors are:
(z - 2i)(z + 2i)
(z^2 + 4)
For the real zeros 2 + 2.236 and 2 - 2.236, the corresponding linear factors are:
(z - (2 + 2.236))(z - (2 - 2.236))
(z - 4.236)(z - 1.764)
To obtain the polynomial function of least degree with integral coefficients, we multiply these factors together:
P(z) = (z^2 + 4)(z - 4.236)(z - 1.764)
Expanding the polynomial:
P(z) = (z^2 + 4)(z^2 - 6z + 7.494
Now, we can multiply the coefficients by a common factor to make them integral.
To do this, we can multiply the entire polynomial by 1000 to eliminate the decimal point:
P(z) = 1000(z^2 + 4)(z^2 - 6z + 7.494)
Expanding the polynomial:
P(z) = 1000z^4 - 6000z^3 + 7494z^2 + 4000z - 23976
Therefore, the polynomial function of least degree with integral coefficients whose zeros are 2i, -2i, 2 + 2.236, and 2 - 2.236 is:
P(z) = 1000z^4 - 6000z^3 + 7494z^2 + 4000z - 23976