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

POLYNOMIAL FUNCTIONS

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 ?

POLYNOMIAL FUNCTIONS

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

Polynomial functions with a degree of 2 are called QUADRATIC POLYNOMIAL FUNCTIONS

Polynomial functions with a degree of 3 are called CUBIC POLYNOMIAL FUNCTIONS

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

POLYNOMIAL FUNCTIONS

EVALUATING A POLYNOMIAL FUNCTION

Find f(2a) if f(x) = 3x2 – 2x – 6

f(2a) = 3(2a)2 – 2(2a) – 6

f(2a) = 12a2 – 4a – 6

POLYNOMIAL FUNCTIONS

EVALUATING A POLYNOMIAL FUNCTION

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

POLYNOMIAL FUNCTIONS

EVALUATING A POLYNOMIAL FUNCTION

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

POLYNOMIAL FUNCTIONS

GENERAL SHAPES OF POLYNOMIAL FUNCTIONS

f(x) = 3

Constant Function

Degree = 0

Max. Zeros: 0

POLYNOMIAL FUNCTIONS

GENERAL SHAPES OF POLYNOMIAL FUNCTIONS

f(x) = x + 2

LinearFunction

Degree = 1

Max. Zeros: 1

POLYNOMIAL FUNCTIONS

GENERAL SHAPES OF POLYNOMIAL FUNCTIONS

f(x) = x2 + 3x + 2

QuadraticFunction

Degree = 2

Max. Zeros: 2

POLYNOMIAL FUNCTIONS

GENERAL SHAPES OF POLYNOMIAL FUNCTIONS

f(x) = x3 + 4x2 + 2

CubicFunction

Degree = 3

Max. Zeros: 3

POLYNOMIAL FUNCTIONS

GENERAL SHAPES OF POLYNOMIAL FUNCTIONS

f(x) = x4 + 4x3 – 2x – 1

QuarticFunction

Degree = 4

Max. Zeros: 4

POLYNOMIAL FUNCTIONS

GENERAL SHAPES OF POLYNOMIAL FUNCTIONS

f(x) = x5 + 4x4 – 2x3 – 4x2 + x – 1

QuinticFunction

Degree = 5

Max. Zeros: 5

POLYNOMIAL FUNCTIONS

END BEHAVIOR

Degree: Even

Leading Coefficient: +

End Behavior:

As x -∞; f(x) +∞

As x +∞; f(x) +∞

f(x) = x2

POLYNOMIAL FUNCTIONS

END BEHAVIOR

Degree: Even

Leading Coefficient: –

End Behavior:

As x -∞; f(x) -∞

As x +∞; f(x) -∞

f(x) = -x2

POLYNOMIAL FUNCTIONS

END BEHAVIOR

Degree: Odd

Leading Coefficient: +

End Behavior:

As x -∞; f(x) -∞

As x +∞; f(x) +∞

f(x) = x3

POLYNOMIAL FUNCTIONS

END BEHAVIOR

Degree: Odd

Leading Coefficient: –

End Behavior:

As x -∞; f(x) +∞

As x +∞; f(x) -∞

f(x) = -x3

REMAINDER AND FACTOR THEOREMS

f(x) = 2x2 – 3x + 4

2 2 -3 4

2 4 1

2 6

Divide the polynomial by x – 2

Find f(2)

f(2) = 2(2)2 – 3(2) + 4f(2) = 8 – 6 + 4f(2) = 6

REMAINDER AND FACTOR THEOREMS

f(x) = 3x5 – 4x3 + 5x - 3

Find f(-3)

Try this one:Remember – Some terms are missing

When synthetic division is used to evaluate a function, it is called

SYNTHETIC SUBSTITUTION.

REMAINDER AND FACTOR THEOREMS

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

FACTOR THEOREM

REMAINDER AND FACTOR THEOREMS

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

2 1 -3 -4 12

1 2

-1 -2 -6

-12 0

Yes, it is a factor, since f(2) = 0.

Can you find the two remaining factors?

REMAINDER AND FACTOR THEOREMS

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

Find the two unknown ( ? ) quantities.