Final answer:
To express the polynomial g(x) as a product of linear factors with 3 as a zero, first factor out (x - 3), then divide g(x) by this factor to find the remaining factors, and finally express g(x) as the product of all its linear factors.
Step-by-step explanation:
If 3 is a zero of the polynomial g(x) = x³ - 9x² + 24x - 18, we can express g(x) as a product of linear factors by factoring the polynomial. Since 3 is a zero, one of the factors is (x - 3). To find the other factors, we can perform polynomial division or use synthetic division to divide g(x) by (x - 3).
We find that the division results in a quotient of x² - 6x + 6, which decomposes further into linear factors using the quadratic formula or by further factorization techniques if factorable. When we factor fully, we express the original polynomial as a product of its linear factors: g(x) = (x - 3)(ax - b)(cx - d), where a, b, c, and d are constants determined through the factoring process.
For instance, if we find x² - 6x + 6 further factors into (x - 2)(x - 3), then the fully factored form of the polynomial would be g(x) = (x - 3)(x - 2)(x - 3), which simplifies to g(x) = (x - 3)²(x - 2).