A polynomial function is a function of the form

1
A polynomial function is a function of the form
f (x) an x n an 1 x n 1 a 1 x
a 0
Where an ? 0 and the exponents are all whole
numbers.
For this polynomial function, an is the
leading coefficient, a 0 is the constant term,
and n is the degree.
A polynomial function is in standard form if
its terms are written in descending order of
exponents from left to right.
2
You are already familiar with some types of
polynomial functions. Here is a summary of common
types ofpolynomial functions.
0
Constant
f (x) a 0
1
Linear
f (x) a1x a 0
2
Quadratic
f (x) a 2 x 2 a 1 x a 0
3
Cubic
f (x) a 3 x 3 a 2 x 2 a 1 x a 0
4
Quartic
f (x) a4 x 4 a 3 x 3 a 2 x 2 a 1 x a 0
3
Decide whether the function is a polynomial
function. If it is, write the function in
standard form and state its degree, typeand
leading coefficient.
SOLUTION
The function is a polynomial function.
It has degree 4, so it is a quartic function.
The leading coefficient is 3.
4
Decide whether the function is a polynomial
function. If it is, write the function in
standard form and state its degree, typeand
leading coefficient.
SOLUTION
The function is not a polynomial function because
the term 3 x does not have a variable base and
an exponentthat is a whole number.
5
Decide whether the function is a polynomial
function. If it is, write the function in
standard form and state its degree, typeand
leading coefficient.
SOLUTION
The function is not a polynomial function because
the term2x 1 has an exponent that is not a
whole number.
6
Decide whether the function is a polynomial
function. If it is, write the function in
standard form and state its degree, typeand
leading coefficient.
SOLUTION
The function is a polynomial function.
It has degree 2, so it is a quadratic function.
The leading coefficient is ?.
7
Polynomial function?
f (x) x 3 3x
f (x) 6x2 2 x 1 x
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One way to evaluate polynomial functions is to
usedirect substitution. Another way to evaluate
a polynomialis to use synthetic substitution.
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SOLUTION
2 x 4 0 x 3 (8 x 2) 5 x (7)
Polynomial in standard form
2 0 8 5 7




3
Coefficients
6
18
30
105
35
10
98
2
6
The value of f (3) is the last number you
write, In the bottom right-hand corner.
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GRAPHING POLYNOMIAL FUNCTIONS
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GRAPHING POLYNOMIAL FUNCTIONS
12
GRAPHING POLYNOMIAL FUNCTIONS
13
Graph f (x) x 3 x 2 4 x 1.
SOLUTION
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.
14
Graph f (x) x 4 2x 3 2x 2 4x.
SOLUTION
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.
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Assignment