Find a polynomial function of lowest degree with rational coefficients that has the given numbers as some of its zeros.
sqrt of 3, 4
Thanks!
To find a polynomial function with rational coefficients that has the given numbers as its zeros, we need to use the property of conjugate pairs for square roots.
First, let's consider √3 as a zero. Since the coefficients of the polynomial function need to be rational, the conjugate of √3, which is -√3, must also be a zero.
Similarly, let's take 4 as another zero. So, -4 will also be a zero.
To find the polynomial function, we can start by writing it in factored form. The factored form is given by:
f(x) = (x - zero1)(x - zero2)(x - zero3)...
Since we know that the function has two zeros, √3 and 4, the factored form becomes:
f(x) = (x - √3)(x + √3)(x - 4)(x + 4)
We can simplify this expression by multiplying the factors:
f(x) = (x^2 - 3)(x^2 - 16)
Expanding further:
f(x) = x^4 - 19x^2 + 48
Therefore, the polynomial function of lowest degree with rational coefficients that has the given numbers as some of its zeros is f(x) = x^4 - 19x^2 + 48.