(a).

Identify Possible Rational Zeros:

Usually it is not practical to test all possible zeros of a polynomial function using only synthetic substitution. The Rational Zero Theorem can be used for finding the some possible zeros to test.

The function is .

Because the leading coefficient is , the possible rational zeros are the intezer factors of the constant term .

or

Therefore the possible rational zeros of are 

Examine the variations in sign for and :

Consider  .

The original polynomial equation has variations in sign.

.

.

has variations in sign.

Therefore by Descartes rule of signs has either or positive real zeros and or negative real zeros.

Synthetic Division :

Consider  .

Perform the synthetic division method by testing and .

Since ,  conclude that is a zero of .

Therefore, is a rational zero.

The depressed polynomial is .

Consider .

Perform the synthetic division method on the depressed polynomial by testing and .

Since ,  conclude that is a zero of .

Therefore, is a rational zero.

Therefore, and are the factors of .

The remaining factor is can be written as .

The remaining quadratic factor yeilds no real zeros and is therefore irreducible over the reals.

So is written as a product of linear and irreducable quadratic factors is

 (b).

The remaining quadratic factor is does not have rational zeros.

To find the rational zeros use the quadratic formula .

Consider  .

Where .

Substiute the values in the quadratic formula .

.

The rational zeros are and .

So written as a product of linear factors is .

(c).

Because the function has degree , by the corollary of the fundemental therom of algebra has excatly zeros,including any that may be repeated.

The linear factorization yeilds zeros:.

(a).

So is written as a product of linear and irreducable quadratic factors is

.

(b).

written as a product of linear factors is .

(c).

The zeros are .