(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
:
\
\
.
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 division method 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 can be written as
.
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).
\ 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).
\ 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 .