Find a polynomial function of lowest degree with rational coefficients that has the given numbers as some of its zeros.
sqrt of 3 and 4i
disregard the first post, thanks!
since the roots come in conjugate pairs,
y = (x+√3)(x-√3)(x-4i)(x+4i)
= (x^2-3)(x^2+16)
...
To find a polynomial function with rational coefficients that has the given numbers (sqrt(3) and 4i) as some of its zeros, we can use the concept of conjugate roots.
Since the coefficients are rational, the complex zeros must occur in conjugate pairs. Therefore, if 4i is a zero, then its conjugate, -4i, must also be a zero.
Now, let's construct the polynomial function step-by-step:
Step 1: Start by setting up the factors for the polynomial.
Since sqrt(3) is a zero, one of the factors can be (x - sqrt(3)).
Since 4i and -4i are zeros, the corresponding factors would be (x - 4i) and (x + 4i).
Step 2: Multiply the factors to obtain the polynomial function.
(x - sqrt(3))(x - 4i)(x + 4i)
Step 3: Simplify by expanding the expression.
(x - sqrt(3))(x^2 + 16)
= x(x^2 + 16) - sqrt(3)(x^2 + 16)
= x^3 + 16x - x^2sqrt(3) - 16sqrt(3)
So, a polynomial function of lowest degree with rational coefficients that has sqrt(3) and 4i as some of its zeros is:
f(x) = x^3 + (16 - sqrt(3))x - 16sqrt(3)
To find a polynomial function with rational coefficients that has √3 and 4i as zeros, you need to consider their conjugates as well.
The conjugate of √3 is -√3, and the conjugate of 4i is -4i.
So, the polynomial function will have the following zeros:
√3, -√3, 4i, and -4i.
To find the polynomial, you can use the fact that the product of all the factors (x - zero) should be equal to zero.
Therefore, the polynomial function can be calculated as follows:
(x - √3)(x + √3)(x - 4i)(x + 4i)
Now, expand this expression:
(x^2 - (√3)^2)(x^2 - (4i)^2)
Simplifying further:
(x^2 - 3)(x^2 + 16)
Expand again:
x^4 - 3x^2 + 16x^2 - 48
Combine like terms:
x^4 + 13x^2 - 48
So, the polynomial function of lowest degree with rational coefficients that has √3 and 4i as some of its zeros is:
f(x) = x^4 + 13x^2 - 48