1. 5.1 Polynomials and Functions
Chapter 5
5.1 Polynomials and Functions

2. n n – 1 f (x) = an x n + an – 1 x n – 1 +· · ·+ a 1 x + a 0 a 0 a0
A polynomial function is a function of the form
f (x) = an x n + an – 1 x n – 1 +· · ·+ a 1 x + a 0
a 0 a0
constant term
an  0 an
leading coefficient
descending order of exponents from left to right. n
n – 1 n
degree
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.

3. You are already familiar with some types of polynomial
functions. Here is a summary of common types of polynomial functions.
Degree
Type
Standard Form
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

4. Identifying Polynomial Functions
Decide whether the function is a polynomial function. If it is,
write the function in standard form and state its degree, type and leading coefficient.
f (x) = x 2 – 3x4 – 7 1 2
SOLUTION
The function is a polynomial function.
Its standard form is f (x) = – 3x x 2 – 7. 1 2
It has degree 4, so it is a quartic function.
The leading coefficient is – 3.

5. Identifying Polynomial Functions
Decide whether the function is a polynomial function. If it is,
write the function in standard form and state its degree, type and leading coefficient.
f (x) = x x
SOLUTION
The function is not a polynomial function because the term 3 x does not have a variable base and an exponent that is a whole number.

6. Identifying Polynomial Functions
Decide whether the function is a polynomial function. If it is,
write the function in standard form and state its degree, type and leading coefficient.
f (x) = 6x x –1 + x
SOLUTION
The function is not a polynomial function because the term 2x –1 has an exponent that is not a whole number.

7. Identifying Polynomial Functions
Decide whether the function is a polynomial function. If it is,
write the function in standard form and state its degree, type and leading coefficient.
f (x) = – 0.5 x +  x 2 – 2
SOLUTION
The function is a polynomial function.
Its standard form is f (x) =  x2 – 0.5x –
It has degree 2, so it is a quadratic function.
The leading coefficient is .

8. f (x) = x 2 – 3 x 4 – 7 f (x) = x 3 + 3x f (x) = 6x2 + 2 x– 1 + x
Identifying Polynomial Functions
Polynomial function?
f (x) = x 2 – 3 x 4 – 7 1 2
f (x) = x x
f (x) = 6x2 + 2 x– 1 + x
f (x) = – 0.5x +  x2 – 2

9. Decide whether the function is a polynomial function
Decide whether the function is a polynomial function. If it is, write the function in standard form and state the degree and leading coefficient.
Test 1, 2

10. Goal 1: To evaluate polynomial functions
Goal 2: To simplify polynomial functions

13. 5.2 Addition and Subtraction of Polynomials
Chapter 5
5.2 Addition and Subtraction of Polynomials
Goal 1: To add polynomial functions
Goal 2: To subtract polynomial functions

15. The additive inverse of a polynomial.
The additive inverse of a polynomial can be found by replacing each term by its additive inverse.
The sum of a polynomial and its additive inverse is O.

17. The additive inverse of a polynomial.
The additive inverse of a polynomial can be found by replacing each term by its additive inverse.
The sum of a polynomial and its additive inverse is O.
Thus, to subtract one polynomial from another, we add its additive inverse.

20. Simplify the polynomial
Test 1, 2

21. HW #5.1-2 Pg 208-209 1-29 Odd, 30-36 Pg 212-213 1-31 Odd, 33-35

22. HW Quiz HW #5.1-2 Sunday, April 23, 2017Pg Pg Pg Pg Pg Pg

23. 5.3 Multiplication of Polynomials
Chapter 5
5.3 Multiplication of Polynomials

33. Find the product.
Test

34. Simplify.
Challenge

35. Based on your answers to parts to the above, write a general formula
Based on your answers to parts to the above, write a general formula. Use “2n” to represent a general even integer and let “2n + 1” represent a general odd integer, and use “…” for missing terms.

36. Answers to challenge

37. HW #5.3 Pg Odd, 40-49

38. HW Quiz HW #5.3 Sunday, April 23, 2017

39. Missing Parts 5.4 Factoring Do Examples from Regular book la205bad
HW 5.4 Pg Every Third, 61-76
5.5 More Factoring
HW Handout Factoring
5.6 Factoring A General Strategy
Do bonus problems from Great Factoring Problems WS
HW Pg Odd, 38-47

40. HW Quiz HW #5.6 Sunday, April 23, 2017
Row 1, 3, 5
Factor Completely 1. 2. 3. 4.
Row 2, 4, 6
Factor Completely 1. 2. 3. 4.

41. 5.7 Solving by Factoring

44. HW #5.7 Pg 233 1-42 Left Column, 43-46 Pg 228 89-99 Odd

45. 5.8 Using Polynomial Equations

47. A candy factory needs a box that has a volume of 30 cubic inches
A candy factory needs a box that has a volume of 30 cubic inches. The width should be 2 inches less than the height and the length should be 5 inches greater than the height. What should the dimensions of the box be?

48. For the city park commission, you are designing a marble planter in which to plant flowers. You want the length of the planter to be six times the height and the width to be three times the height. The sides should be one foot thick. Since the planter will be on the sidewalk, it does not need a bottom. What should the outer dimensions of the planter be if it is to hold 4 cubic feet of dirt?

49. Suppose you have 250 cubic inches of clay with which to make a rectangular prism for a sculpture. If you want the height and width each to be 5 inches less than the length, what should the dimensions of the prism be?

50. HW #5.8a Pg Odd, 18-20

51. 5 15 17 18 7 13 20

52. Test Review

53. Geometry Express the area A of a rectangle as a function of the length x if the length of the rectangle is twice its width.
Geometry Express the area A of an isosceles right triangle as a function of the length x of one of the two equal sides.

54. Find the degree of a polynomial
Find the degree of a monomial
Find the function value of a polynomial
Standard form of a polynomial
Find volume of a box
Properties of Exponents
Additive Inverse
Polynomial arithmetic
Factoring
Solving Polynomial equations
Study all the challenge problems in the book

58. Evaluate: (30)2 + 2(30)(22) + (22)2

60. Factor a9 b9c9d2 - d2 x2n + xn yn – yn-1 + yn-2 72x2n + 120xn + 50
50x4 – 72y6
x2 – w2 –16 + 8w

61. Factor
x4 + 9x2 + 81
x4 + x2y2 + 25y4
x4 + 64
7y2a + b – 5ya + b + 3ya + 2b
4x2a – 4xa – 3
a6 – 64b6
9a3 +9a2b – 4ab2 – 4b3

62. HW #R-4 Pg