Final answer:
By dividing the cubic polynomial by (x + 2) and using the quadratic formula, we can express the given function as a product of linear factors, if all roots are real, or as a product of one linear factor and an irreducible quadratic factor, if there are complex roots.
Step-by-step explanation:
To express the function f(x) = x³ - 2x² + 5x + 26 as a product of linear factors knowing that -2 is a zero, we perform polynomial division or use synthetic division with the factor (x + 2).
Once we divide the polynomial by (x + 2), we are left with a quadratic equation, which we can solve for the remaining zeros using the quadratic formula, x = (-b ± sqrt(b² - 4ac)) / (2a). After finding these zeros, if they exist, the function can be fully factored into the product of linear factors.
However, if we do not obtain real solutions from the quadratic equation, it means that the polynomial has a pair of complex-conjugate roots. In this case, the polynomial would be expressed as a product of a linear factor and an irreducible quadratic factor.