Answer
a. true
b. true
c. false
Work Step by Step
a.
By the Zeros Theorem (p.289)
"Every polynomial of degree n $\geq$ 1 has exactly n zeros,
provided that a zero of multiplicity k is counted k times."
The degree of P(x) is 4, so, the statement is true.
b.
By the Complete Factorization Theorem (p.287),
" ... $P$ factors into $n$ linear factors : $P(x)=a(x-c_{1})(x-c_{2})\cdots(x-c_{n})$
where $a$ is the leading coefficient of $P$ and $c_{1}, c_{1}, \ldots, c_{n}$ are the zeros of $P$ ",
the statement is true.
c. If there exists a c that is a real zero, then
$c^{4}+1=0$
$c^{4}=-1$
which can not be, as even powers of real numbers are nonnegative.
So, such a c does not exist.
The statement is false.
