The function is .

Draw the coordinate plane.

Graph the function .

Graph :

Observe the graph :

From the graph it appears that the real zeros of this function lies in the interval .

Lower bound is and upper bound is .

Test the lower bound and upper bound .

Every number in the last line is alternately non negative and non positive.

So, is a lower bound.

Every number in the last line is non negative.

So, is a upper bound.

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 .

or

.

Because the real zeros are in the interval we can narrow this list to just

.

The function is .

From the graph it appears that are reasonable.

Perform synthetic substitution method by testing .

The depressed polynomial is .

Perform the synthetic substitution method on the depressed polynomial by testing .

The new depressed polynomial is .

Perform the synthetic substitution method on the new depressed polynomial by testing .

The new depressed polynomial is .

Perform the synthetic substitution method on the new depressed polynomial by testing

.

Therefore, are the factors of .

By using Factor theorem,

When  then   is a factor of polynomial.

Factoring of .

Zeros are .

So has five real zeros and all are rational.

The graph supports the conclusion.

The possible rational zeros are

.

The zeros of are .