(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 .

Therefore the possible rational zeros of are .

Examine the variations in sign for and :

Consider .

The original polynomial equation has variations in sign.

.

.

There are no sign changes.so ,there are no negative roots.

Therefore by Descartes rule of signs has either or positive real zeros and no negative real zeros.

Synthetic Division:

Consider .

Setup the synthetic division using a zero place for the missing term term in the dividend.

.

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 substitution division 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 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 is written as a product of linear and irreducable quadratic factors is

.

(b).

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).

The function 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 .