Download Section 5.1 - Shelton State

Section 3.1
Polynomial Functions and
Models
Polynomial Functions
A polynomial of degree n is a function of the form
P(x) = anxn + an-1xn-1 + ... + a1x + a0

Where an
0. The numbers a0, a1, a2, . . . , an are
called the coefficients of the polynomial. The a0 is
the constant coefficient or constant term. The
number an, the coefficient of the highest power, is
the leading coefficient, and the term anxn is the
leading term.
Example of a Polynomial Function
P( x)  4 x  2 x  5x  3
4
3
Graphs of Polynomial Functions and
Nonpolynomial Functions
Graphs of Polynomials
• Graphs are lines
– Degree 0 or 1
ex. f(x) = 3 or f(x) = x – 5
• Graphs are parabolas
– Degree 2
ex. f(x) = x2 + 4x + 8
• Graphs are smooth
curves
– Degree greater than 2
ex. f(x) = x3
• These graphs will not
have the following:
– Break or hole
– Corner or cusp
End Behavior of Polynomials
End Behavior- a description of what
happens as x becomes large in the
positive and negative direction.
End Behavior is determined by:
• Term with the highest power of x
• Sign of this term’s coefficient
Even- and Odd-Degree Functions
The Leading-Term Test
Finding Zeros of a Polynomial
Zero- another way of saying solution
Zeros of Polynomials
• Solutions
• Place where graph crosses the x-axis
(x-intercepts)
• Zeros of the function
Place where f(x) = 0
X-Intercepts (Real Zeros)
• A polynomial function of degree n will
have at most n x-intercepts (real
zeros).
Number of Turning Points
(relative maxima/minima)
The number of relative maxima/minima
of the graph of a polynomial function of
degree n is at most n – 1.
ex. f(x) = x4 + 3x3 – 2x2 + 1
Determine number of relative
maxima/minima
n – 1 = 4 – 1 = 3
Using the Graphing Calculator
to Determine Zeros
Graph the following polynomial function and determine the zeros.
P( x)  x  5x  x  21x  18
4
3
2
Before graphing, determine the end behavior and the number
of relative maxima/minima.
In factored form:
P(x) = (x + 2)(x – 1)(x – 3)²
Multiplicity
If (x-c)k, k  1, is a factor of a polynomial
function P(x) and:
K is odd
– The graph crosses the
x-axis at (c, 0)
K is even
– The graph is tangent to
the x-axis at (c, 0)
Multiplicity
y = (x + 2)²(x − 1)³
Answer.
−2 is a root of multiplicity 2,
and 1 is a root of multiplicity 3.
These are the 5 roots:
−2, −2, 1, 1, 1.
Multiplicity
y = x³(x + 2)4(x − 3)5
Answer.
0 is a root of multiplicity 3,
-2 is a root of multiplicity 4,
and 3 is a root of multiplicity 5.
True or False?
• 1.) The function
P( x )  x  3 x  2 x  5
3
2
must
have 1 real zero.
• 2.) The function
P( x )  3 x  5
4
has no real zeros.
• 3.) An odd degree polynomial function must have at
least 1 real zero.
• 4.) An even degree polynomial function must have at
least 1 real zero.
To Graph a Polynomial
1. Use the leading term to determine the end
behavior.
2. Find all its real zeros (x-intercepts).
Set y = 0.
3. Use the x-intercepts to divide the graph into
intervals and choose a test point in each
interval to graph.
4. Find the y-intercept. Set x = 0.
5. Use any additional information (i.e. turning
points or multiplicity) to graph the function.
The Intermediate Value Theorem
Consider a polynomial function P(x) with the
points (a, P(a)) and (b, P(b)) on the function.
For any P(x) with real coefficients, suppose
that for a ≠ b, P(a) and P(b) are of opposite
signs. Then the function has a real zero
between a and b.
The Intermediate Value Theorem
In other words, if one point is above
the x-axis and the other point is
below the x-axis, then because P(x) is
continuous and will have to cross the
x-axis to connect the two points, P(x)
must have a zero somewhere between
a and b.