Section 16
Polynomial Function is a function of the form P(x) = anxn + … + a1x + a0, where an  0,
n  0 and n is called the degree of the function. a0, a1, …, an are called the coefficients
of the polynomial. an is the leading coefficient, anxn is the leading term, and a0 is the
constant term or constant coefficient.

The graph of a polynomial function is a smooth curve. A graph that is continuous
and has no sharp points is smooth.
Leading Coefficient Test: As x moves without bound to the left or the right, the graph
of the polynomial function f(x) = anxn + …+ a1x + a0 eventually rises or falls in the
following manner.
1. When n is odd:

If the leading coefficient is positive (an > 0), the graph falls to the left and rises to
the right.

If the leading coefficient is negative (an < 0), the graph rises to the left and falls to
the right.
2. When n is even:

If the leading coefficient is positive (an > 0), the graph rises to the left and rises to
the right.

If the leading coefficient is negative (an < 0), the graph falls to the left and falls to
the right.
Zeros of Polynomial Functions: It can be shown that for a polynomial function f of
degree n, the following statements are true:

The graph of f has, at most, n – 1 turning points. (Turning points are points at which
the graph changes from increasing or decreasing or vice versa.)

The function f has, at most, n real zeros.
Real Zeros of a Polynomial Functions: If f is a polynomial function and a is a real
number, the following statements are equivalent:

x = a is a zero of the function f.

x = a is a solution of the polynomial equation f(x) = 0.

(x – a) is a factor of the polynomial f(x).

(a, 0) is an x – intercept of the graph of f.
Repeated Zeros: A factor (x – a)k, k > 1, yields a repeated zero x = a of multiplicity k.

If k is odd, the graph crosses the x – axis at x = a.

If k is even, the graph touches the x – axis at x = a.
EX: Find all the real zeros of the polynomial function. Determine the multiplicity of each
zero.
a. f(t) = t3 – 4t2 + 4t
b. f(x) = 5x4 + 15x2 + 10
EX: Sketch the graph of the function described and determine a function that satisfies the
conditions described. The leading term is –x5, the graphs only zeros are x = -3, 2, and 4.
Also the multiplicity of the zero at x = -3 is 3.
End Behavior of the Graph:
1. A polynomial with even-degree and a positive leading coefficient has end behavior
that as x goes to -  , y goes to  and as x goes to  , y goes to  .
2. A polynomial with even-degree and a negative leading coefficient has end behavior
that as x goes to -  , y goes to -  and as x goes to  , y goes to -  .
3. A polynomial with odd-degree and a positive leading coefficient has end behavior
that as x goes to -  , y goes to -  and as x goes to  , y goes to  .
4. A polynomial with odd-degree and a negative leading coefficient has end behavior
that as x goes to -  , y goes to  and as x goes to  , y goes to -  .
EX: Sketch the graph of the function described and determine a function that satisfies the
conditions described. The function has a y – intercept at (0,0), has factors x – 3, x + 5,
and x with multiplicities 1, 3, and 2 respectively. The end behavior of the graph is as x
goes to -  , y goes to -  and as x goes to  , y goes to -  .