Download Chapter 3 Polynomial Functions 3.1 Polynomials

Chapter 3 Polynomial Functions
3.1 Polynomials
Objectives:
A.APR.1: Understanding that polynomials form a system analogous to the integers, namely
they are closed under the operations of addition, subtraction, and multiplication;
add, subtract and multiply polynomials.
A.CED.2: Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
For the board: You will be able to identify, evaluate, add and subtract polynomials.
You will be able to classify and graph polynomials.
Bell Work 3.1:
Evaluate.
1. -24
Simplify each expression.
3. – 2(3x – 1)
2. (-2)4
4. 3(y2 + 6y)
Anticipatory Set:
A monomial is a number or a product of numbers and variables with whole number exponents.
Examples: 24, xyz, 4a2b3
A polynomial is a monomial or a sum or difference of monomials.
Each monomial in a polynomial is a term.
Example: 5x3 – 3x2 – 4x + 8
The degree of a monomial is the sum of the exponents of the variables.
Example: 4a2b3 has degree 5.
Open the book to page 150 and read example 1.
Example: identify the degree of each monomial.
a. z6
b. 5.6
c. 8xy3
6
0
4
d. a2bc
4
White Board Activity:
Practice: Identify the degree of each monomial.
a. x3
b. 7
c. 5x3y2
3
0
5
d. a6bc2
9
A polynomial with two terms is called a binomial.
A polynomial with three terms is called a trinomial.
Example: 3a - b
Example: x2 – 3x - 4
Polynomials should be written in standard form or descending order.
If the polynomial contains multiple variables, choose one of the variables to write in descending
order. This will put the term with the highest power of the chosen variable first. Its coefficient
is known as the leading coefficient.
The degree of a polynomial is the degree of the term with the highest degree.
Name
Constant
Linear
Quadratic
Cubic
Quartic
Quintic
Classifying Polynomials by Degree
Degree
Example
0
-9
1
x–4
2
2
x + 3x – 1
3
3
x + 2x2 + x + 1
4
2x4 + x3 + 3x2 + 4x – 1
5
7x5 + x4 – x3 + 3x2 + 2x - 1
Open the book to page 151 and read example 2.
Example: Rewrite each polynomial in standard form. Then identify the leading coefficient, degree, and
number of terms. Name the polynomial.
a. 3 – 5x2 + 4x
b. 3x2 – 4 + 8x4
– 5x2 + 4x + 3
8x4 + 3x2 – 4
leading coefficient: -5
leading coefficient: 8
degree: 2
degree: 4
name: quadratic trinomial
name: quartic trinomial
White Board Activity:
Practice: Rewrite each polynomial in standard form. Then identify the leading coefficient, degree, and
number of terms. Name the polynomial.
a. 4x – 2x2 + 2
b. -18x2 + x3 – 5 + 2x
– 2x2 + 4x + 2
x3 - 18x2 + 2x – 5
Leading coefficient: -2
leading coefficient: 1
Degree: 2
degree: 3
Name: quadratic trinomial
name: cubic polynomial
Open the book to page 151 - 152 and read example 3.
Example: Add or subtract. Write your answer in standard form.
a. (2x3 + 9 – x) + (5x2 + 4 + 7x + x3)
b. (3 – 2x2) – (x2 + 6 – x)
3
2x
– x+ 9
-2x2
+3
x3 + 5x2 + 7x + 4
- x2 + x – 6
3x3 + 5x2 + 6x + 13
-3x2 + x – 3
White Board Activity:
Practice: Add or subtract. Write your answer in standard form.
a. (-36x2 + 6x – 11) + (6x2 + 16x3 – 5)
b. (5x3 + 12 + 6x2) – (15x2 + 3x – 2)
- 36x2 + 6x – 11
5x3 + 6x2
+ 12
3
2
2
16x + 6x
– 5
- 15x - 3x + 2
3
2
3
16x – 30x + 6x – 16
5x – 9x2 – 3x + 14
A polynomial function is a function whose rule is a polynomial.
Open the book to page 152 and read example 4.
Example: The volume of a box is modeled by the function V(h) = h3 + 3h2 + 2h, where h represents
the height of the box in inches. Evaluate V(3) and V(10) and describe what the values
represent.
V(3) = 33 + 3 ∙ 32 + 2 ∙ 10 = 27 + 27 + 6 = 60 in3
The volume of the box is 60 in3 when the height of the box is 3 in.
V(10) = 103 + 3 ∙ 102 + 2 ∙ 10 = 1000 + 300 + 20 = 1320 in3
The volume of the box is 1320 in3 when the height of the box is 10 in.
White Board Activity:
Practice: The volume of a pyramid is modeled by the function V(h) = 3h 3 + 9h2 + 6h, where h represents
the height of the pyramid in feet. Evaluate V(3) and V(30) and describe what the values
represent.
V(3) = 3 ∙ 33 + 9 ∙ 32 + 6 ∙ 3 = 81 + 81 + 18 = 180 ft3
The volume of the pyramid is 180 ft3 when the height is 3 ft.
V(30) = 3 ∙ 303 + 9 ∙ 302 + 6 ∙ 30 = 81,000 + 8,100 + 180 = 89,280
The volume of the pyramid is 89,280 ft3 when the height is 30 ft.
Assessment:
Question student pairs.
Independent Practice:
Text: pg. 154 – 155 prob. 2 – 14, 19 – 31, 36 – 39.
For a grade:
Text: pg. 154 – 155 prob. 14, 20, 24, 30,