(a)

The polynomial function is and is a one factor of f.

Rewrite the expression in long division form .

The number zero is included for missing term x in the divisor.

The remainder is the last entry in the last row, so, .

The number along the bottom row are the coefficients of the quotient.

By the division algorithm theorem, the  result of the division is .

The  factorization (factors are irreducible over the rationals) of f(x) is

(b)

The  factorization (factors are irreducible over the rationals) of f(x) is

Factor the expression .

.

The product of linear and quadratic factors that are irreducible over the reals is .

(c)

Consider the quadratic factor and solve by using quadratic formula.

The  complete factorization of f(x) is .

(a)

The  factorization (factors are irreducible over the rationals) of f(x) is

(b)

The product of linear and quadratic factors that are irreducible over the reals is .

(c)

The  complete factorization of f(x) is .