(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 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 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 .

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).

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

. Then factor the differences of squares as .

So, written as a product of linear factor 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).

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

.

(b).

written as a product of linear factor as follows .

(c).

The zeros are .