(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.

has variation in sign.

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

Synthetic Division :

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 factor is .

We can factor by writing the expression first as a difference of squares .

Then factor this difference of squares as and .

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

.

(b).

written as a product of linear factor ia as follows .

(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 factor ia as follows .

(c).

The zeros are .