Math 150 – Fall 2015
Section 5A
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Section 5A – Polynomial Functions
Definition. A polynomial function is any function p(x) of the form
p(x) = pn xn + pn−1 xn−1 + · · · + p2 x2 + p1 x + p0
where all of the exponents are non-negative integers and pn 6= 0.
• The degree of the polynomial is n.
• The coefficient p0 is called the constant term, and the pn is called the leading
coefficient.
Example 1.
polynomial
degree
constant term
leading coefficient
p(x) = 4x7 − 3x5 − 6x + 11
p(x) = 3x − 5 + 9x22 − 3x11
p(x) = 8
Behavior of Polynomial Functions
Definition. For a function f (x), the phrase “as x → ∞, f (x) → ∞” means that as
x goes to infinity (becomes arbitrarily large), the function values f (x) go to infinity
(become arbitrarily large).
Theorem. Let p(x) = pn xn + pn−1 xn−1 + · · · + p2 x2 + p1 x + p0 with pn 6= 0, and the
degree n is odd.
f (x) = x3
If pn > 0,
then as x → ∞, p(x) → ∞
and as x → −∞, p(x) → −∞.
If pn < 0,
then as x → ∞, p(x) → −∞
and as x → −∞, p(x) → ∞.
f (x) = −x3
Math 150 – Fall 2015
Section 5A
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Theorem. Let p(x) = pn xn + pn−1 xn−1 + · · · + p2 x2 + p1 x + p0 with pn 6= 0, and the
degree n is even.
f (x) = x2
If pn > 0,
then as x → ∞, p(x) → ∞
and as x → −∞, p(x) → ∞.
f (x) = −x2
If pn < 0,
then as x → ∞, p(x) → −∞
and as x → −∞, p(x) → −∞.
Note. The end behavior for p(x) = pn xn +pn−1 xn−1 +· · ·+p2 x2 +p1 x+p0 is determined
by pn xn only. All other terms can be ignored! So we just need to determine if pn is
positive or negative, and as x → ∞ or −∞ is xn positive or negative. Then we will
know if the product pn xn is going to ∞ or −∞.
• If n is even, then xn is always positive.
• If n is odd, then xn has the same sign as x.
Example 2. Describe the end behavior of the polynomials below.
(a) p(x) = 5x − 7x37 − 11x41 − 12x111
(b) p(x) = (3x5 − 4x)(8x17 − 5x6 )
(c) p(x) = (−3x7 − 5x)(8x + 10x11 )
Example 3. If p(x) is a polynomial of degree 113 and the values of p(x) go to ∞ as x
goes to ∞, then what happens to p(x) for large negative values of x?