C H A P T E R 2 Polynomial and Rational Functions

Section 2.2 Polynomial Functions of Higher Degree
Section 2.2 Polynomial Functions of Higher Degree 151
You should know the following basic principles about polynomials.
■ is a polynomial function of degree n.
■ If f is of odd degree and
(a) then (b) then
1. 1.
2. 2.
■ If f is of even degree and
(a) then (b) then
1. 1.
2. 2.
■ The following are equivalent for a polynomial function.
(a) is a zero of a function.
(b) is a solution of the polynomial equation
(c) is a factor of the polynomial.
(d) is an x-intercept of the graph of f.
■ A polynomial of degree n has at most n distinct zeros and at most turning points.
■ A factor x a yields a repeated zero of x a of multiplicity k.
(a) If k is odd, the graph crosses the x-axis at x a.
(b) If k is even, the graph just touches the x-axis at x a.
■ If f is a polynomial function such that a < b and fa fb, then f takes on every value between fa and fb in the
interval a, b.
■ If you can find a value where a polynomial is positive and another value where it is negative, then there is at least one real
zero between the values.
k fx anx an > 0,
an < 0,
fx → as x → .
fx → as x → .
fx → as x → .
fx → as x → .
an > 0,
an < 0,
fx → as x → .
fx → as x → .
fx → as x → .
fx → as x → .
x a
x a
fx 0.
x a
a, 0
n 1
, k > 1,
n an1xn1 . . . a2x2 a1x a0 , an 0,
Vocabulary Check
1. continuous 2. Leading Coefficient Test 3.
4. solution; x a; x-intercept
5. touches; crosses 6. standard
7. Intermediate Value
n; n 1
1. fx 2x 3 is a line with y-intercept 0, 3.
2. fx x is a parabola with intercepts 0, 0 and
Matches graph (c).
4, 0 and opens upward. Matches graph (g).
2 4x
3. is a parabola with x-intercepts
0, 0 and and opens downward. Matches
graph (h).
5
fx 2x
2 , 0
2 5x 4. has intercepts
and Matches graph (f).
1 1
2 23, 0.
1
fx 2x 0, 1, 1, 0,
1
2 23, 0
3 3x 1
5. fx has intercepts 0, 0 and ±23, 0.
Matches graph (a).
1
4x4 3x2 6. fx has y-intercept
Matches graph (e).
1
3x3 x2 4
3
0, 4
3.
7. fx x has intercepts 0, 0 and 2, 0.
Matches graph (d).
4 2x3 8. fx has intercepts 0, 0, 1, 0,
1, 0, 3, 0, 3, 0. Matches graph (b).
1
5x5 2x3 9
5x