Find a polynomial function of lowest degree with rational coefficients that has the given numbers as some of its zeros.

2-i,sqrt2

f(x)= ?

Thanks!

since the coefficients are real, complex roots occur in conjugate pairs. So, 2+i is another root. Similarly, since the coefficients are rational, we need to balance that √2 with -√2.

f(x) = (x-(2-i))(x-(2+i))(x-√2)(x-(-√2))
= ((x-2)+i)((x-2)-i)(x-√2)(x+√2)
= ((x-2)^2+1)(x^2-2)
Now all the i's and √2's are gone
...

To find a polynomial function with the given zeros, we can use the fact that complex conjugates occur in pairs and use the rational root theorem.

Since 2-i is a zero, the complex conjugate 2+i is also a zero.
Since √2 is also a zero, the conjugate -√2 is also a zero.

Using these zeros, we can write the polynomial function as follows:

f(x) = (x - 2 + i)(x - 2 - i)(x - √2)(x + √2)

Simplifying further:

f(x) = ((x - 2)^2 - i^2)(x^2 - 2)

Since i^2 = -1:

f(x) = ((x - 2)^2 + 1)(x^2 - 2)

Multiplying and simplifying:

f(x) = (x^2 - 4x + 5)(x^2 - 2)

Expanding:

f(x) = x^4 - 2x^2 - 4x^3 + 8x + 5x^2 - 10

Combining like terms:

f(x) = x^4 - 4x^3 + 6x^2 + 8x - 10

Therefore, the polynomial function of lowest degree with rational coefficients that has the zeros 2-i, 2+i, √2, and -√2 is:

f(x) = x^4 - 4x^3 + 6x^2 + 8x - 10.

To find a polynomial function with rational coefficients that has the given numbers as some of its zeros, we'll use the concept of conjugate pairs.

First, let's consider the zero 2 - i. Since complex zeros come in conjugate pairs, we know that the conjugate of 2 - i is 2 + i, which must also be a zero of the polynomial.

Now, let's consider the zero √2. Since the square root of 2 is irrational, we cannot directly use this as a coefficient in the polynomial to obtain rational coefficients. However, we can use the fact that if a polynomial has irrational zeros, then their conjugates must also be zeros.

So, the conjugate of √2 is -√2, which must also be a zero of the polynomial.

Therefore, the polynomial function with the given zeros is found by multiplying the factors of (x - (2 - i)), (x - (2 + i)), and (x - √2)(x - (-√2)):

(x - (2 - i))(x - (2 + i))(x - √2)(x + √2)

Now, let's simplify this expression:

(x - 2 + i)(x - 2 - i)(x - √2)(x + √2)

Expanding the first two binomials using the difference of squares:

((x - 2)^2 - (i)^2)(x - √2)(x + √2)

Simplifying the squares of binomials:

(x^2 - 4x + 4 + 1)(x - √2)(x + √2)

Combining like terms:

(x^2 - 4x + 5)(x^2 - 2)

Multiplying the binomials:

x^4 - 6x^2 + 8x - 10

Therefore, the polynomial function with the given zeros is:

f(x) = x^4 - 6x^2 + 8x - 10