Final answer:
To write h(x) as the product of linear factors and find its zeros, various methods like the rational root theorem can be applied to determine rational zeros; the polynomial is then factored accordingly.
Step-by-step explanation:
To write the polynomial h(x) = x^3 – 5x^2 + 8x - 6 as the product of linear factors and to find all the zeros of the function, one may need to apply methods such as factoring by grouping, synthetic division, or using the rational root theorem. However, since this particular polynomial does not factor easily by inspection, we can try to find rational roots using the rational root theorem, which suggests that any rational root of this polynomial, in its most reduced form, will be a factor of the constant term (6) over a factor of the leading coefficient (1).
After testing possible rational zeros using synthetic division or by evaluating h(x) for these possible values, suppose we have found a zero of the polynomial, let's call it 'a'. Once a zero is found, we use synthetic division to factor out (x - a) and get a quadratic factor, which can be factored further or solved using the quadratic formula. This process will yield the linear factors and the remaining zeros of the polynomial.
The exact factors and zeros cannot be given without further information or clear steps that led to finding a particular zero.