A polynomial function is known to satisfy the following properties: • The graph is…

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

Problem

A polynomial function

g

is known to satisfy the following properties: • The graph is symmetric about the origin. • It intersects the

x

–axis at

x=-2,0,2

and no other points. • As

x\to+\infty

, the values of

g(x)

are negative. Answer the following questions. (i) State the least possible degree of

g

, giving reasons. (ii) Construct one possible algebraic expression for

g(x)

with integer coefficients that fits all the above properties and has leading coefficient

-1

. (iii) Verify that your expression in part (ii) indeed produces the stated

x

-intercepts.