(a) To factor the polynomial P(x) = x^3 - 4x^2 + 4x - 16 into linear and irreducible quadratic factors with real coefficients:
We notice that the polynomial is not divisible by any common factor. We can then proceed to factor by grouping or using other methods. In this case, we'll use the rational root theorem to find the roots and then factor accordingly.
The rational root theorem states that any rational root of a polynomial equation in the form P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_0 must be of the form ± p/q, where p is a factor of the constant term a_0 and q is a factor of the leading coefficient a_n.
For our polynomial P(x) = x^3 - 4x^2 + 4x - 16, the constant term is -16 and the leading coefficient is 1. The factors of -16 are ±1, ±2, ±4, ±8, and ±16, and the factors of 1 are ±1.
By testing these factors as potential rational roots using synthetic division or polynomial long division, we find that x = 4 is a root of the polynomial.
Performing synthetic division with x = 4, we get:
P(x) = (x - 4)(x^2 - 16)
The quadratic factor x^2 - 16 is a difference of squares, so it factors further into (x + 4)(x - 4).
Therefore, the factored form of P(x) into linear and irreducible quadratic factors with real coefficients is:
P(x) = (x - 4)(x + 4)(x - 4)
(b) To factor P(x) completely into linear factors with complex coefficients, we use the factored form from part (a) and further factor the quadratic factor x^2 - 16.
P(x) = (x - 4)(x + 4)(x - 4)
The quadratic factor x^2 - 16 factors into (x + 4)(x - 4).
Therefore, the completely factored form of P(x) into linear factors with complex coefficients is:
P(x) = (x - 4)(x + 4)(x - 4)
