Objectives Identify, evaluate, add, and subtract polynomials. Classify and graph polynomials.

Define the following (from the back of the textbook): monomial polynomial degree of a monomial degree of a polynomial leading coefficient binomial trinomial polynomial function

A monomial is a number or a product of numbers and variables with whole number exponents. A polynomial is a monomial or a sum or difference of monomials. Polynomials have no variables in denominators or exponents, no roots or absolute values of variables, and all variables have whole number exponents. 1 2 a7 Polynomials: 3x4 2z12 + 9z3 0.15x101 3t2 – t3 8 5y2 1 2 Not polynomials: 3x |2b3 – 6b| m0.75 – m The degree of a monomial is the sum of the exponents of the variables.

Example 1: Identifying the Degree of a Monomial Identify the degree of each monomial. A. z6 B. 5.6 C. 8xy3 D. a2bc3

Degree of a polynomial is given by the term with the greatest degree. Standard Form Terms are written in descending order by degree. -The degree of the first term indicates the degree of the polynomial. - Leading coefficient is the coefficient of the first term.

A polynomial can be classified by its number of terms: A polynomial with one term is a monomial. A polynomial with two terms is called a binomial. A polynomial with three terms is called a trinomial.

A polynomial can also be classified by its degree:

Example 2: Classifying Polynomials 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 Leading coefficient: Leading coefficient: Degree: Degree: Terms: Terms: Name: Name:

Your Turn: Example 2 Rewrite each polynomial in standard form. Then identify the leading coefficient, degree, and number of terms. Name the polynomial. c. 4x – 2x2 + 2 d. –18x2 + x3 – 5 + 2x Leading coefficient: Leading coefficient: Degree: Degree: Terms: Terms: Name: Name:

Exit Slip/Homework: Find the degree of the monomials. 1. 4 𝑎 2 𝑏 𝑐 7 2. 4 𝑥 3 Find the degree of the polynomials. 3. 3𝑥 2 +4𝑥−1 4. 3𝑥 6 −5 𝑥 8 +4 𝑥 7 Write in standard form. 5. 2𝑥− 4𝑥 4 −10+3 𝑥 2 −6 𝑥 5

To ADD or SUBTRACT polynomials, combine like terms!!

Example 3: Adding and Subtracting Polynomials Add or subtract. Write your answer in standard form. A. (2x3 + 9 – x) + (5x2 + 4 + 7x + x3)

Example 3: Adding and Subtracting Polynomials Add or subtract. Write your answer in standard form. B. (3 – 2x2) – (x2 + 6 – x)

Your Turn: Example 3a Add or subtract. Write your answer in standard form. c. (–36x2 + 6x – 11) + (6x2 + 16x3 – 5)

Your Turn: Example 3b Add or subtract. Write your answer in standard form. d. (5x3 + 12 + 6x2) – (15x2 + 3x – 2)

Exit Slip/Homework 1. 3 𝑥 2 +2𝑥−5 +( 6𝑥 2 −5𝑥+7) 1. 3 𝑥 2 +2𝑥−5 +( 6𝑥 2 −5𝑥+7) 2. 𝑥 5 + 4𝑥 3 +8𝑥−5 +( 4𝑥 4 +2 𝑥 3 −2𝑥−1) 3. 6𝑥−5 −( 3𝑥 2 +𝑥−3) 4. 5 𝑥 3 −7 𝑥 2 +3𝑥+9 −( 2𝑥 3 +4 𝑥 2 −𝑥)

6-2 Multiplying Polynomials Warm Up: Use the properties of exponents to simplify the following terms. 𝑎 4 ∙ 𝑎 5 (𝑓 3 ) 5 ( 𝑝 3 𝑞 𝑟 4 ) 2

Objectives 6-2 Multiplying Polynomials Multiply polynomials. Use binomial expansion to expand binomial expressions that are raised to positive integer powers.

6-2 Multiplying Polynomials To multiply a polynomial by a monomial, use the Distributive Property and the Properties of Exponents.

Example 1: Multiplying a Monomial and a Polynomial 6-2 Multiplying Polynomials Example 1: Multiplying a Monomial and a Polynomial Find each product. A. 4y2(y2 + 3) B. fg(f4 + 2f3g – 3f2g2 + fg3)

6-2 Multiplying Polynomials Your Turn: Example 1 Find each product. a. 3cd2(4c2d – 6cd + 14cd2) b. x2y(6y3 + y2 – 28y + 30)

Example 2A: Multiplying Polynomials Find the product. (a – 3)(2 – 5a + a2)

Example 2B: Multiplying Polynomials Find the product. (y2 – 7y + 5)(y2 – y – 3) y2 –y –3 y2 –7y 5 y4 –y3 –3y2 –7y3 7y2 21y 5y2 –5y –15

6-2 Multiplying Polynomials Check It Out! Example 2a Find the product. (3b – 2c)(3b2 – bc – 2c2)

6-2 Multiplying Polynomials Check It Out! Example 2b Find the product. (x2 – 4x + 1)(x2 + 5x – 2) x2 –4x 1 x2 5x –2 x4 –4x3 x2 5x3 –20x2 5x –2x2 8x –2

Warm Up Write the polynomial in standard form. Then identify the leading coefficient, the degree and number of terms. 2𝑥− 4𝑥 4 −10+3 𝑥 2 −6 𝑥 5 a. standard form: ______________ b. leading coefficient: ___________ c. degree of polynomial: _________ d. number of terms: ____________ e. name: _____________________ 2. 6𝑥−5 − 3𝑥 2 +𝑥−3 3. ( 2𝑥 2 −4𝑥+3)( 𝑥 2 −3𝑥+7)

6-2 Multiplying Polynomials Check It Out! Example 4a Find the product. (x + 4)4

6-2 Multiplying Polynomials Check It Out! Example 4b Find the product. (2x – 1)3 8x3 – 12x2 + 6x – 1

Example 5: Using Pascal’s Triangle to Expand Binomial Expressions 6-2 Multiplying Polynomials Example 5: Using Pascal’s Triangle to Expand Binomial Expressions Expand each expression. A. (k – 5)3 B. (6m – 8)3

6-2 Multiplying Polynomials Check It Out! Example 5 Expand each expression. a. (x + 2)3 x3 + 6x2 + 12x + 8 b. (x – 4)5 x5 – 20x4 + 160x3 – 640x2 + 1280x – 1024

6-2 Multiplying Polynomials Check It Out! Example 5 Expand the expression. c. (3x + 1)4 81x4 + 108x3 + 54x2 + 12x + 1

Exit Slip/Homework: Find the sum/difference. Find the product. 1. 2 𝑥 2 +4𝑥−6 +(3𝑥+7) 2. 𝑥 5 −3 𝑥 3 +2𝑥 −(7 𝑥 4 +5 𝑥 3 −10) Find the product. 3. 5𝑥 3 ( 2𝑥 2 +3𝑥−1) 4. ( 2𝑥 2 −4𝑥+3)( 𝑥 2 −3𝑥+7)

6-2 Multiplying Polynomials Lesson Quiz Find each product. 1. 5jk(k – 2j) 5jk2 – 10j2k 2. (2a3 – a + 3)(a2 + 3a – 5) 2a5 + 6a4 – 11a3 + 14a – 15 3. The number of items is modeled by 0.3x2 + 0.1x + 2, and the cost per item is modeled by g(x) = –0.1x2 – 0.3x + 5. Write a polynomial c(x) that can be used to model the total cost. –0.03x4 – 0.1x3 + 1.27x2 – 0.1x + 10 4. Find the product. (y – 5)4 y4 – 20y3 + 150y2 – 500y + 625 5. Expand the expression. (3a – b)3 27a3 – 27a2b + 9ab2 – b3