Final answer:
To use the remainder theorem, set up the equation P(a) = 0 and solve for a by trying different values. By factoring P(x) completely, we get (x - 2)(x - 1)(x - 3).
Step-by-step explanation:
To use the remainder theorem to factor the polynomial P(x) = x^3 - 6x^2 + 11x - 6, we need to find the values of x that make P(x) equal to zero. The remainder theorem states that if a polynomial P(x) is divided by the expression x - a and the remainder is zero, then (x - a) is a factor of P(x). So, we set up the equation P(a) = 0, and solve for a by trying different values of a. By factoring P(x) completely, we get (x - 2)(x - 1)(x - 3), which gives us the completely factored form of P(x).