2. A polynomial function is a function of the form
f (x) = an x nn + an – 1 x nn––1 1 · ·+ a 1 x + a 0 a 0
+·
n
0
Where an ≠ 0 and the exponents are all whole numbers.
n
For this polynomial function, an is the
an
constant term
a
a 00 is the constant term, and n is the
n
leading coefficient
leading coefficient,
degree
degree.
A polynomial function is in standard form if its terms are
descending order of exponents from left to right.
written in descending order of exponents from left to right.
3. You are already familiar with some types of polynomial
functions. Here is a summary of common types of
polynomial functions.
Degree
Type
Standard Form
0
Constant
f (x) = a
1
Linear
f (x) = a1x + a
2
Quadratic
f (x) = a 2 x 2 + a 1 x + a
3
Cubic
f (x) = a 3 x 3 + a 2 x 2 + a 1 x + a
4
Quartic
f (x) = a4 x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a
4. Identifying Polynomial Functions
Decide whether the function is a polynomial function. If it is,
write the function in standard form and state its degree, type
and leading coefficient.
f (x) =
1 2
x – 3x4 – 7
2
S OLUTION
The function is a polynomial function.
Its standard form is f (x) = – 3x 4 +
1 2
x – 7.
2
It has degree 4, so it is a quartic function.
The leading coefficient is – 3.
5. Identifying Polynomial Functions
Decide whether the function is a polynomial function. If it is,
write the function in standard form and state its degree, type
and leading coefficient.
f (x) = x 3 + 3 x
S OLUTION
The function is not a polynomial function because the
x
term 3 does not have a variable base and an exponent
that is a whole number.
6. Identifying Polynomial Functions
Decide whether the function is a polynomial function. If it is,
write the function in standard form and state its degree, type
and leading coefficient.
–
f (x) = 6x 2 + 2 x 1 + x
S OLUTION
The function is not a polynomial function because the term
2x –1 has an exponent that is not a whole number.
7. Identifying Polynomial Functions
Decide whether the function is a polynomial function. If it is,
write the function in standard form and state its degree, type
and leading coefficient.
f (x) = – 0.5 x + π x 2 –
2
S OLUTION
The function is a polynomial function.
Its standard form is f (x) = π x2 – 0.5x – 2.
It has degree 2, so it is a quadratic function.
The leading coefficient is π.
9. Using Synthetic Substitution
One way to evaluate polynomial functions is to use
direct substitution.
Use substitution to evaluate
Use substitution to evaluate
f (x) = 2 x 4 + −8 x 2 + 5 x − 7 when x = 3.
10. Now use direct substitution:
f (x) = 2 x 4 + −8 x 2 + 5 x − 7 when x = 3.
f ( x) = 2(3) 4 − 8(3) 2 + 5(3) − 7
= 98
11. Using Synthetic Substitution
One way to evaluate polynomial functions is to use
direct substitution. Another way to evaluate a polynomial
is to use synthetic substitution.
Use synthetic division to evaluate
f (x) = 2 x 4 + −8 x 2 + 5 x − 7 when x = 3.
12. Using Synthetic Substitution
Start by writing it in
standard form
S OLUTION
2 x 4 + 0 x 3 + (–8 x 2) + 5 x + (–7)
Polynomial
Polynomial inin
standard form
standard form
3•
3
2
0
–8
5
–7
Coefficients
6
18
30
105
6
10
35
98
x-value
2
The value of (3) is the last number you write,
The value of ff(3) is the last number you write,
In the bottom right-hand corner.
In the bottom right-hand corner.
17. If “n” is even, the graph of the polynomial is “U-shaped”
meaning it is parabolic (the higher the degree, the
more curves the graph will have in it).
If “n” is odd, the graph of the polynomial is “snake-like”
meaning looks like a snake (the higher the degree, the
more curves the graph will have in it).
19. Leading Coefficient Test
Degree is odd
Leading
coefficient
is positive
Leading
coefficient
is negative
Degree is even
Start low,
End high
Leading
coefficient
is positive
Start high,
End high
Start high,
End low
Leading
coefficient
is negative
Start low,
End low
23. Determine the left and right behavior of the graph
of each polynomial function.
f(x) = x4 + 2x2 – 3x
f(x) = -x5 +3x4 – x
f(x) = 2x3 – 3x2 + 5
24. Determine the left and right behavior of the graph
of each polynomial function.
f(x) = x4 + 2x2 – 3x
Even, Leading coefficient 1 (positive) , starts high
ends high
f(x) = -x5 +3x4 – x
Odd, Leading coefficient 1 (negative) , starts
high ends low
f(x) = 2x3 – 3x2 + 5
ODD , Leading coefficient 2 (positive) , starts
LOW ends HIGH
25. Tell me what you know about the
equation…
Odd / Even ?
Leading coefficient Positive or Negative?
26. Tell me what you know about the
equation…
Odd / Even ?
Leading coefficient Positive or Negative?
27. Tell me what you know about the
equation…
Odd / Even ?
Leading coefficient Positive or Negative?
31. G RAPHING P OLYNOMIAL F UNCTIONS
The end behavior of a polynomial function’s graph
is the behavior of the graph as x approaches infinity
(+ ∞) or negative infinity (– ∞). The expression
x
+ ∞ is read as “x approaches positive infinity.”
33. Graphing Polynomial Functions
Graph f (x) = x 3 + x 2 – 4 x – 1.
S OLUTION
To graph the function, make a table of
values and plot the corresponding points.
Connect the points with a smooth curve
and check the end behavior.
x
f(x)
–3
–2
–1
0
1
2
3
34. Graphing Polynomial Functions
Graph f (x) = x 3 + x 2 – 4 x – 1.
S OLUTION
To graph the function, make a table of
values and plot the corresponding points.
Connect the points with a smooth curve
and check the end behavior.
x
f(x)
–3
–7
–2
3
–1
3
0
–1
1
–3
2
3
3
23
35. Graphing Polynomial Functions
Graph f (x) = –x 4 – 2x 3 + 2x 2 + 4x.
S OLUTION
To graph the function, make a table of
values and plot the corresponding points.
Connect the points with a smooth curve
and check the end behavior.
x
f (x)
–3
–2
–1
0
1
2
3
36. Graphing Polynomial Functions
Graph f (x) = –x 4 – 2x 3 + 2x 2 + 4x.
S OLUTION
To graph the function, make a table of
values and plot the corresponding points.
Connect the points with a smooth curve
and check the end behavior.
x
f (x)
–3
–21
–2
0
–1
–1
0
0
1
3
2
–16
3
–105
