Final answer:
The polynomial f(x) = x^3 - 7x - 6, with 3 as a zero, can be factored into linear terms by first dividing by (x-3) and then using the quadratic formula to solve the resulting quadratic equation for its zeros.
Step-by-step explanation:
To express the polynomial f(x) = x³ - 7x - 6 as a product of linear factors, given that 3 is a zero, we first divide the polynomial by (x-3) using synthetic division or long division. Once we have factored out (x-3), the remaining quadratic can be factored further or solved for its zeros using the quadratic formula. The quadratic formula states that for any equation of the form ax² + bx + c = 0, the solutions for x are given by x = (-b ± √(b² - 4ac)) / (2a). After finding the two remaining zeros, we can rewrite the polynomial as the product of three linear factors.
For example, if after dividing by (x-3) we obtain a quadratic equation like x² + 0.0211x - 0.0211 = 0, we would apply the quadratic formula to find the remaining zeros, which then gives us the other two linear factors.