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. Complex Numbers Note that squaring both sides yields: therefore and so
And so on…

19. Real numbers and imaginary numbers are subsets of the set of complex numbers.

20. Definition of a Complex Number
If a and b are real numbers, the number a + bi is a complex number, and it is said to be written in standard form.
If b = 0, the number a + bi = a is a real number.
If a = 0, the number a + bi is called an imaginary number.
Write the complex number in standard form

21. Addition and Subtraction of Complex Numbers
If a + bi and c +di are two complex numbers written in standard form, their sum and difference are defined as follows.
Sum:
Difference:

22. Perform the subtraction and write the answer in standard form.
( 3 + 2i ) – ( i )
3 + 2i – 6 – 13i
–3 – 11i 4

23. Multiplying Complex Numbers
Multiplying complex numbers is similar to multiplying polynomials and combining like terms.
Perform the operation and write the result in standard form. ( 6 – 2i )( 2 – 3i )
F O I L
12 – 18i – 4i + 6i2
12 – 22i + 6 ( -1 )
6 – 22i

24. The Fundamental Theorem of Algebra
We have seen that if a polynomial equation is of degree n, then counting multiple roots separately, the equation has n roots. This result is called the Fundamental Theorem of Algebra.
The Fundamental Theorem of Algebra
If f (x) is a polynomial of degree n, where n  1, then the equation f (x) = 0 has at least one complex root.

25. The Linear Factorization Theorem
Just as an nth-degree polynomial equation has n roots, an nth-degree polynomial has n linear factors. This is formally stated as the Linear Factorization Theorem.
The Linear Factorization Theorem
If f (x) = anxn + an-1xn-1 + … + a1x + a0 b, where n  1 and an  0 , then
f (x) = an (x - c1) (x - c2) … (x - cn)
where c1, c2,…, cn are complex numbers (possibly real and not necessarily distinct). In words: An nth-degree polynomial can be expressed as the product of n linear factors.

26. Find all the zeros of
Solutions:
The possible rational zeros are
Synthetic division or the graph can help:
Notice the real zeros appear as x-intercepts. x = 1 is repeated zero since it only “touches” the x-axis, but “crosses” at the zero x = -2.
Thus 1, 1, and –2 are real zeros. Find the remaining 2 complex zeros.

27. Write a polynomial function f of least degree that has real coefficients, a leading coefficient 1, and 2 and 1 + i as zeros).
Solution:
f(x) = (x – 2)[x – (1 + i)][x – (1 – i)]

28. Factoring Cubic Polynomials

29. Find the Greatest Common Factor
Identify each term in the polynomial.
14x3 – 21x2
2•7•x•x•x
3•7•x•x –
Identify the common factors in each term
GCF = 7x2
The GCF is?
Use the distributive property to factor out the GCF from each term
14x3 – 21x2 =
7x2(2x – 3)

30. Factor Completely 4x3 + 20x2 + 24x 2•2•x•x•x 2•2•5•x•x 2•2•2•3•x
Identify each term in the polynomial.
4x3 + 20x2 + 24x
Identify the common factors in each term
2•2•x•x•x
2•2•5•x•x
2•2•2•3•x + +
GCF = 4x
The GCF is?
4x3 + 20x2 + 24x =
4x(x2 + 5x +6)
Use the distributive property to factor out the GCF from each term 4x
(x + 2)(x + 3)

31. Factor by Grouping x3 - 2x2 - 9x + 18 = (x3 - 2x2) + (- 9x + 18) +
Group terms in the polynomial.
= (x3 - 2x2) + (- 9x + 18)
Identify a common factor in each group and factor +
x•x•x-2•x•x
-3•3•x+2•3•3
= x2(x – 2) +
-9(x – 2)
Now identify the common factor in each term
= (x – 2)(x2 – 9)
Use the distributive property
= (x – 2)(x – 3)(x + 3)
Factor the difference of two squares

32. Sum of Two Cubes Pattern
a3 + b3 = (a + b)(a2 - ab + b2)
Example
Now, use the pattern to factor
x =
x3 + 3•3•3
= x3 + 33
x = (x + 3)(x2 - 3x + 32)
= (x + 3)(x2 - 3x + 9)
So x = (x + 3)(x2 - 3x + 9)

33. Difference of Two Cubes Pattern
a3 - b3 = (a - b)(a2 + ab + b2)
Example
Now, use the pattern to factor
n =
n3 - 4•4•4
= n3 - 43
n = (n - 4)(n2 + 4n + 42)
= (n - 4)(n2 + 4n + 16)
So n = (n - 4)(n2 + 4n + 16)