(a).

The polynomial functions are:

(1) .

The degree is , therefore will have zeros.

(2) .

The degree is , therefore will have zeros.

(3) .

The degree is , therefore will have zeros.

(4) .

The degree is ,Therefore will have zeros.

(5) .

The degree is ,Therefore will have zeros.

(6) .

The degree is ,Therefore will have zeros.

(b).

Identify Possible Rational Zeros:

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. 

(1).

.

Because the leading coefficient is , the possible rational zeros are the intezer factors of the constant term .

Therefore the possible rational zeros of are .

Perform synthetic division by testing .

Therefore it is determined that is a rational zero.

The remaining quadratic factor can be written as can be written as .

Therefore the zeros of are .

(2).

.

Replace missing terms with zero coeficient.

Because the leading coefficient is , the possible rational zeros are the intezer factors of the constant term .

or .

Therefore the possible rational zeros of are .

Perform synthetic division by testing .

Therefore it is determined that is a rational zero.

Perform synthetic division on the depressed polynomial by testing .

Therefore it is determined that is a rational zero.

Perform synthetic division on the new depressed polynomial by testing .

The remaining quadratic factor is which can be written as and it can be written as .

Therefore the zeros of are .

(3).

.

Because the leading coefficient is , the possible rational zeros are the intezer factors of the constant term .

or .

Therefore the possible rational zeros of are .

Perform synthetic division by testing .

Therefore it is determined that is a rational zero.

The remaining quadratic factor is which can be written as .

Therefore the zeros of are .

(4).

.

Substiute and factor .

The factor can be written as and the factor can be written as

.

Therefore the zeros of are .

(5).

.

can be written as .

Because the leading coefficient is , the possible rational zeros are the intezer factors of the constant term .

or

.

Therefore the possible rational zeros of are

.

Perform synthetic division by testing .

Therefore it is determined that is a rational zero.

Perform synthetic division on the depressed polynomial by testing .

The remaining quadratic factor is which can be written as .

can be written as .

The zeros of are

(6).

.

Because the leading coefficient is , the possible rational zeros are the intezer factors of the constant term .

or .

Perform synthetic division by testing .

Therefore it is determined that is a rational zero.

Perform synthetic division on the depressed polynomial by testing .

The remaining quadratic factor is which can be written as .

can be written as .

The zeros of are .

(c).

An odd-degree polynomial function always has an odd number of zeros and a polynomial function with real coefficients has imaginary zeros that occur in conjugate pairs.

Therefore, an odd function with real coefficients will always have at least one real zero.

(a).

(1)  The degree is ,Therefore will have zeros.

(2) The degree is ,Therefore will have zeros.

(3) The degree is ,Therefore will have zeros.

(4) The degree is ,Therefore will have zeros.

(5) The degree is ,Therefore will have zeros.

(6) The degree is ,Therefore will have zeros.

(b).

(1)The zeros of are .

(2)The zeros of are .

(3)The zeros of are .

(4)The zeros of are .

(5)The zeros of are

(6)The zeros of are .

(c).

An odd function with real coefficients will always have at least one real zero.