Polynomial functions

POLYNOMIAL FUNCTIONS

A POLYNOMIAL is a monomial
or a sum of monomials.

A POLYNOMIAL IN ONE
VARIABLE is a polynomial that
contains only one variable.
Example: 5x2 + 3x - 7

The DEGREE of a polynomial in one variable is
the greatest exponent of its variable.

A LEADING COEFFICIENT is the coefficient
of the term with the highest degree.

What is the degree and leading
coefficient of 3x5 – 3x + 2 ?

A polynomial equation used to represent a
function is called a POLYNOMIAL FUNCTION.
Polynomial functions with a degree of 1 are called
LINEAR POLYNOMIAL FUNCTIONS

QUADRATIC POLYNOMIAL FUNCTIONS
CUBIC POLYNOMIAL FUNCTIONS

EVALUATING A POLYNOMIAL FUNCTION

Find f(-2) if f(x) = 3x2 – 2x – 6

f(-2) = 3(-2)2 – 2(-2) – 6
f(-2) = 12 + 4 – 6
f(-2) = 10


Find f(2a) if f(x) = 3x2 – 2x – 6

f(2a) = 3(2a)2 – 2(2a) – 6
f(2a) = 12a2 – 4a – 6


Find f(m + 2) if f(x) = 3x2 – 2x – 6
f(m + 2) = 3(m + 2)2 – 2(m + 2) – 6

f(m + 2) = 3(m2 + 4m + 4) – 2(m + 2) – 6
f(m + 2) = 3m2 + 12m + 12 – 2m – 4 – 6
f(m + 2) = 3m2 + 10m + 2


Find 2g(-2a) if g(x) = 3x2 – 2x – 6
2g(-2a) = 2[3(-2a)2 – 2(-2a) – 6]

2g(-2a) = 2[12a2 + 4a – 6]
2g(-2a) = 24a2 + 8a – 12

GENERAL SHAPES OF

f(x) = 3
Constant
Function
Degree = 0

Max. Zeros: 0

GENERAL SHAPES OF

f(x) = x + 2
Linear
Function
Degree = 1
Max. Zeros: 1

GENERAL SHAPES OF

f(x) = x2 + 3x + 2
Quadratic
Function
Degree = 2
Max. Zeros: 2

GENERAL SHAPES OF

f(x) = x3 + 4x2 + 2
Cubic
Function
Degree = 3
Max. Zeros: 3

GENERAL SHAPES OF

f(x) = x4 + 4x3 – 2x – 1
Quartic
Function
Degree = 4
Max. Zeros: 4

GENERAL SHAPES OF
f(x) = x5 + 4x4 – 2x3 – 4x2 + x – 1

Quintic
Function
Degree = 5
Max. Zeros: 5

END BEHAVIOR

f(x) = x2
Degree: Even
Leading Coefficient: +
End Behavior:
As x  -∞; f(x)  +∞
As x  +∞; f(x)  +∞

END BEHAVIOR

f(x) = -x2
Degree: Even
Leading Coefficient: –
End Behavior:
As x  -∞; f(x)  -∞
As x  +∞; f(x)  -∞

END BEHAVIOR

f(x) = x3
Degree: Odd
Leading Coefficient: +
End Behavior:
As x  -∞; f(x)  -∞
As x  +∞; f(x)  +∞

END BEHAVIOR

f(x) = -x3
Degree: Odd
Leading Coefficient: –
End Behavior:
As x  -∞; f(x)  +∞
As x  +∞; f(x)  -∞

REMAINDER AND FACTOR
THEOREMS

f(x) = 2x2 – 3x + 4
Divide the polynomial by x – 2
Find f(2)
2 2 -3 4 f(2) = 2(2)2 – 3(2) + 4
4 2
f(2) = 8 – 6 + 4
2 1 6 f(2) = 6

THEOREMS
When synthetic division is used to
evaluate a function, it is called
SYNTHETIC SUBSTITUTION.
Try this one:
Remember – Some terms are missing
f(x) = 3x5 – 4x3 + 5x - 3
Find f(-3)

THEOREMS

FACTOR THEOREM
The binomial x – a is a factor of the
polynomial f(x) if and only if f(a) = 0.

THEOREMS

Is x – 2 a factor of x3 – 3x2 – 4x + 12

2 1 -3 -4 12
-2 -12 Yes, it is a factor,
2
since f(2) = 0.
1 -1 -6 0

Can you find the two remaining factors?

THEOREMS

(x + 3)( ? )( ? ) = x3 – x2 – 17x - 15

Find the two unknown ( ? ) quantities.

Polynomial functions

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