The polynomial contains real parameters .

MATHEMATICS · Multi-part · AP · AP Precalculus · Polynomial and Rational Functions · Polynomial and Rational Functions

Problem

The polynomial

Q(x)=x^{6}+a x^{4}+b x^{2}+c x+d

contains real parameters

a,\,b,\,c,\,d

. (i) Derive the necessary and sufficient conditions on

a,\,b,\,c,\,d

for which

Q(x)

is an even function. (ii) When the conditions in part (i) hold, write

Q(x)

in its simplest form and state its degree and leading coefficient. (iii) Taking

a=-5

and

b=3

(with the conditions from part (i) satisfied), describe the sign of

Q(x)

x\to\infty

and as

x\to-\infty

, and comment on the general shape of its graph.