Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros 2, 2i, and 4-sqrt 6
If you want rational coefficients, then that √6 has to go, just like the i terms. So, you need
f(x) = (x-2)(x-2i)(x+2i)(x-(4-√6))(x-(4+√6))
= (x-2)(x^2+4)((x-4)+√6)((x-4)-√6)
= (x-2)(x^2+4)(x^2-8x+10)
To find a polynomial function with the given zeros, we can use the fact that complex roots occur in conjugate pairs. Thus, if 2i is a root, then its conjugate -2i must also be a root.
So, the zeros are: 2, 2i, -2i, and 4 - sqrt(6).
We can start by writing out the factors for each of these zeros:
(x - 2), (x - 2i), (x + 2i), (x - (4 - sqrt(6))).
Now, we can simplify the last factor:
(x - (4 - sqrt(6))) = (x - 4 + sqrt(6)).
To find the equation with a leading coefficient of 1, we multiply all the factors together:
(x - 2)(x - 2i)(x + 2i)(x - 4 + sqrt(6)).
Multiplying these factors out, we get:
(x^2 - 2x - 2ix + 4i)(x^2 - (4 - sqrt(6))x - 4ix + (8 - 4sqrt(6))).
Simplifying further, we have:
(x^2 - 2x - 2ix + 4i)(x^2 - 4x + sqrt(6)x - 4ix + 8 - 4sqrt(6)).
Expanding this expression, we get:
x^4 - 6x^3 + (9 - 2sqrt(6))x^2 + (-2 - 4sqrt(6))x + (16 - 16sqrt(6)).
Therefore, the polynomial function f(x) with the given zeros is:
f(x) = x^4 - 6x^3 + (9 - 2sqrt(6))x^2 + (-2 - 4sqrt(6))x + (16 - 16sqrt(6)).
To find a polynomial function with the given zeros, we need to consider both the real and imaginary zeros.
First, let's consider the real zero, which is 2. Since this is a zero, it means that when we plug in 2 into our polynomial, it should equal zero. Therefore, we can use the factor theorem to write a factor of our polynomial. The factor will be (x - 2) since (2 - 2) is zero.
Next, let's consider the complex zero, which is 2i. Since complex zeros always occur in conjugate pairs, the other zero will be the conjugate of 2i, which is -2i. Therefore, the two factors for the complex zeros will be (x - 2i) and (x + 2i).
Finally, let's consider the real zero, which is 4 - sqrt(6). Similarly to the first step, we can use the factor theorem to write a factor of our polynomial. The factor will be (x - (4 - sqrt(6))). Simplifying this, we get (x - 4 + sqrt(6)).
Now, to find the polynomial function, we can multiply all the factors together.
f(x) = (x - 2)(x - 2i)(x + 2i)(x - 4 + sqrt(6))
Expanding this multiplication, we get:
f(x) = (x - 2)(x^2 - (2i)^2)(x - 4 + sqrt(6))
f(x) = (x - 2)(x^2 + 4)(x - 4 + sqrt(6))
To simplify further, we can expand the multiplication:
f(x) = (x^3 + 6x^2 - 8x - 32 + 4sqrt(6)x - 24sqrt(6))
Therefore, the polynomial function of least degree with rational coefficients, a leading coefficient of 1, and zeros 2, 2i, and 4 - sqrt(6) is:
f(x) = x^3 + 6x^2 - 8x - 32 + 4sqrt(6)x - 24sqrt(6)