Answer
(a) According to Descartes' Rule, polynomial $P$ has 3 sign changes
which means that it has 3 or 1 positive real zeros.
$P(-x)=2x^4+3x^3-x-15$ and there is one sign change which means
that it has one negative real zeros. In summary, the function $P$ may
has 4, or 2 real zeros.
(b) An upper bound for the zeros of a polynomial?
It means that the maximum of the real zeros will not exceed $b$ and
the minimum of the real zeros will not be less than $a$.
(c) As shown with the synthetic divisions, 3 is an upper bound because
all the coefficients on the quotient remainder line are positive, while
-3 is a low bound because the coefficients on the quotient line
alternates between positive and negative.
Work Step by Step
(a) Explain how Descartes’ Rule of Signs is used to determine
the possible number of positive and negative real roots of P.
According to Descartes' Rule, polynomial $P$ has 3 sign changes
which means that it has 3 or 1 positive real zeros.
$P(-x)=2x^4+3x^3-x-15$ and there is one sign change which means
that it has one negative real zeros. In summary, the function $P$ may
has 4, or 2 real zeros.
(b) What does it mean to say that a is a lower bound and b is
an upper bound for the zeros of a polynomial?
It means that the maximum of the real zeros will not exceed $b$ and
the minimum of the real zeros will not be less than $a$.
(c) Explain how the Upper and Lower Bounds Theorem is
used to show that all the real zeros of P lie between -3 and 3.
As shown with the synthetic divisions, 3 is an upper bound because
all the coefficients on the quotient remainder line are positive, while
-3 is a low bound because the coefficients on the quotient line
alternates between positive and negative.
