\

Step 1:

\

\

\

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 intiger factors of the constant term $\"\"$ . \ \

\

Therefore the possible rational zeros of $\"\"$ are $\"\"$

\

Step 2 :

\

\

\

The function is $\"\"$ .

\

\

Synthetic Division:

\

\

\

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

\

\

$\"\"$

\

Since $\"\"$ , conclude that $\"\"$ is a zero of $\"\"$ .

\

Therefore, $\"\"$ is a rational zero.

\

The depressed polynomial is $\"\"$ .

\

Step 3 :

\

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.

\

The new depressed polynomial is $\"\"$ .

\

Step 4 :

\

Consider the new depressed polynomial $\"\"$ .

\

$\"\"$

\

Therefore $\"\"$ and $\"\"$ are the factors of $\"\"$ .

\

$\"\"$

\

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

\

$\"\"$

\

$\"\"$

\

\

\

Solution :

\

\

The possible rational zeros are $\"\"$ .

\

The zeros of $\"\"$ are $\"\"$ .