1. A POLYNOMIAL is a monomial or a sum of monomials.
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

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

3. 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

4. EVALUATING A POLYNOMIAL FUNCTION
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) = – 6
f(-2) = 10

5. EVALUATING A POLYNOMIAL FUNCTION
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

6. EVALUATING A POLYNOMIAL FUNCTION Find f(m + 2) if f(x) = 3x2 – 2x – 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

7. EVALUATING A POLYNOMIAL FUNCTION Find 2g(-2a) if g(x) = 3x2 – 2x – 6
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

8. POLYNOMIAL FUNCTIONS f(x) = 3 Constant Function Degree = 0
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = 3
Constant Function
Degree = 0
Max. Zeros: 0

9. POLYNOMIAL FUNCTIONS f(x) = x + 2 Linear Function Degree = 1
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x + 2
Linear
Function
Degree = 1
Max. Zeros: 1

10. POLYNOMIAL FUNCTIONS f(x) = x2 + 3x + 2 Quadratic Function Degree = 2
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x2 + 3x + 2
Quadratic
Function
Degree = 2
Max. Zeros: 2

11. POLYNOMIAL FUNCTIONS f(x) = x3 + 4x2 + 2 Cubic Function Degree = 3
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x3 + 4x2 + 2
Cubic
Function
Degree = 3
Max. Zeros: 3

12. POLYNOMIAL FUNCTIONS f(x) = x4 + 4x3 – 2x – 1 Quartic Function
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x4 + 4x3 – 2x – 1
Quartic
Function
Degree = 4
Max. Zeros: 4

13. POLYNOMIAL FUNCTIONS f(x) = x5 + 4x4 – 2x3 – 4x2 + x – 1 Quintic
GENERAL SHAPES OF
POLYNOMIAL FUNCTIONS
f(x) = x5 + 4x4 – 2x3 – 4x2 + x – 1
Quintic
Function
Degree = 5
Max. Zeros: 5

14. POLYNOMIAL FUNCTIONS f(x) = x2 Degree: Even Leading Coefficient: +
END BEHAVIOR
f(x) = x2
Degree: Even
Leading Coefficient: +
End Behavior:
As x  -∞; f(x)  +∞
As x  +∞; f(x)  +∞

15. POLYNOMIAL FUNCTIONS f(x) = -x2 Degree: Even Leading Coefficient: –
END BEHAVIOR
f(x) = -x2
Degree: Even
Leading Coefficient: –
End Behavior:
As x  -∞; f(x)  -∞
As x  +∞; f(x)  -∞

16. POLYNOMIAL FUNCTIONS f(x) = x3 Degree: Odd Leading Coefficient: +
END BEHAVIOR
f(x) = x3
Degree: Odd
Leading Coefficient: +
End Behavior:
As x  -∞; f(x)  -∞
As x  +∞; f(x)  +∞

17. POLYNOMIAL FUNCTIONS f(x) = -x3 Degree: Odd Leading Coefficient: –
END BEHAVIOR
f(x) = -x3
Degree: Odd
Leading Coefficient: –
End Behavior:
As x  -∞; f(x)  +∞
As x  +∞; f(x)  -∞

18. REMAINDER AND FACTOR THEOREMS
f(x) = 2x2 – 3x + 4
Divide the polynomial by x – 2
Find f(2) 2
f(2) = 2(2)2 – 3(2) + 4 4 2
f(2) = 8 – 6 + 4 2 1 6
f(2) = 6

19. REMAINDER AND FACTOR 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)

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

21. REMAINDER AND FACTOR THEOREMS
Is x – 2 a factor of x3 – 3x2 – 4x + 12 2
Yes, it is a factor, since f(2) = 0. 2 -2
-12 1 -1 -6
Can you find the two remaining factors?

22. REMAINDER AND FACTOR THEOREMS Find the two unknown ( ? ) quantities.
(x + 3)( ? )( ? ) = x3 – x2 – 17x - 15
Find the two unknown ( ? ) quantities.