Question

Write the polynomial function of the least degree with integer coefficients given roots \(x = 4 - \sqrt{2}\) and \(x = i\sqrt{3}\) in standard form.

a) \(f(x) = x^2 - 4\sqrt{2}x + 3\)
b) \(f(x) = x^3 - 4x^2\sqrt{2} - 3\)
c) \(f(x) = x^2 - 4x\sqrt{2} - 3\)
d) \(f(x) = x^3 - 4\sqrt{2}x^2 - 3\)

Answer 1

To find a polynomial function with integer coefficients given the roots (x = 4 - sqrt(2}) and (x = isqrt(3}), we can use the fact that complex conjugate roots come in pairs. We can write the polynomial function in factored form and then expand and simplify it to get the final answer.

Step-by-step explanation:

To find a polynomial function with integer coefficients given the roots (x = 4 - sqrt(2}) and (x = isqrt(3}), we can use the fact that complex conjugate roots come in pairs. So, if (x = 4 - sqrt(2}) is a root, then (x = 4 + sqrt(2}) must also be a root. Similarly, if (x = isqrt(3}) is a root, then (x = -isqrt(3}) must also be a root.

Now we can write the polynomial function in factored form as: (x - (4 - sqrt(2}))(x - (4 + sqrt(2}))(x - isqrt(3}))(x - (-isqrt(3})). Multiplying out these factors gives us the polynomial function:

f(x) = (x - 4 + sqrt(2}))(x - 4 - sqrt(2}))(x - isqrt(3}))(x + isqrt(3})

Expanding this expression will give the polynomial function with integer coefficients, and simplifying it will give us the final answer.