91Ó°ÊÓ

To begin factorizing the given polynomial, we might use synthetic division and the Rational Root theorem. The Rational Root theorem states that any rational root of the polynomial \(P(x)\), expressed in its simplest form, can be written as a factor of the constant term divided by a factor of the leading coefficient. The leading coefficient in our case is 3 and the constant term is -24. Their factors are ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24 and ±1, ±3 respectively. Upon inspection, we see that p/q gives us possible rational roots as ±1, ±2, ±3, ±4, ±6, ±8, ± 1/3, ±2/3, ±4/3, ±8/3. By plugging in these values back into the polynomial, we will eventually find the roots and divide the polynomial using synthetic division to further factorize the polynomial.

Thus, by utilizing synthetic division and/or other polynomial factorization methods, we obtain \( ( x - 3 ) ( x + 1 ) ( 3x^2 - 2 ) ( x^2 + 1 )^2 \) as a factorized form of the given polynomial \( P(x) \). Setting these to zero in order to find the roots of the polynomial, we get \( x = 3, -1, \sqrt{2}/3 , -\sqrt{2}/3 , i, -i \). Each of these are the zeroes of the polynomial.

Upon examination of the factorized polynomial, we see that the zeroes \(x = i\) and \(x = -i\) appears twice (are squared in the factorized form). These are thus zeros of multiplicity 2. All other zeroes only appear once, so they are zeroes of multiplicity 1.