The function f, of, x, equals, 4, x, to the power 4 , plus, 23, x, cubed, plus, 43, x, squared, plus, 33, x, plus, 9f(x)=4x 4 +23x 3 +43x 2 +33x+9 has at least two rational roots. Use the rational root theorem to find those roots, then proceed to find all complex roots. (Note: roots may be integer, rational, irrational, and/or complex.)

The Rational Root Theorem states that any rational root, p/q (where p is a factor of the constant term and q is a factor of the leading coefficient), of a polynomial equation

f(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_2x^2 + a_1x + a_0 = 0

must be such that p is a factor of a_0 and q is a factor of a_n.

In this case, the polynomial is f(x) = 4x^4 + 23x^3 + 43x^2 + 33x + 9.

The factors of the constant term (9) are ±1, ±3, ±9. The factors of the leading coefficient (4) are ±1, ±2, ±4.

So, the possible rational roots of the polynomial are ±1, ±3, ±9, ±1/2, ±3/2, ±9/2, ±1/4, ±3/4, ±9/4.

We can use synthetic division or direct substitution to test these possible roots.

After testing these possible roots, we find that -1/2 and -3/2 are the rational roots of the polynomial.

To find the remaining roots, we can now reduce the polynomial using synthetic division by the found roots -1/2 and -3/2. The reduced polynomial will be a quadratic, which we can solve using the quadratic formula.

The quadratic formula is x = [-b ± sqrt(b^2 - 4ac)] / (2a).

After applying the quadratic formula, we find the remaining roots of the polynomial, which may be real or complex.