(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 variation 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: 

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

By The synthetic division method it is determined that has no rational zeros.

To find the factors  of Substiute  .

.

.

.

To find the rational zeros use the quadratic formula .

Consider .

Where .

Substiute the values in the quadratic formula .

The rational roots are and .

Earlier we assume that .

Now and .

Therefore and are the factors of .

The factor can be written as .

The factor has no real zeros and is therefore irreducable 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 difference of squares as and .

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 .