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1. 3. 2. = y 13 = -10d 7 = – 72a 33 b 14

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Algebra 1 ~ Chapter 8.4 Polynomials

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Remember: A monomial is a number, a variable, or a product of numbers and variables with whole-number exponents. “Mono” – single term The degree of a monomial is the sum of the exponents of the variables. A constant has degree 0.

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Ex. 1 - Find the degree of each monomial. A. 4p4q34p4q3 The degree is 7. Add the exponents of the variables: 4 + 3 = 7. B. 7ed A variable written without an exponent has an exponent of 1. 1+ 1 = 2. C. 3 There is no variable, but you can write 3 as 3x 0. The degree is 2. The degree is 0.

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* A polynomial is the sum or difference of monomials. The degree of a polynomial is the degree of the term with the greatest degree. “poly” – many An example of a polynomial is 3a + 4b – 8c That expression consists of three monomials “combined” with addition or subtraction.

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Some polynomials have special names based on the number of terms they have.

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Ex. 2 – Find the degree of each polynomials. Then name the polynomials based on # of terms. A.) 5m 4 + 3m B.) -4x 3 y 2 + 3x 2 + 5 C.) 3a + 7ab – 2a 2 b The greatest degree is 4, so the degree of the polynomial is 4. The degree of the polynomial is 5. The degree of the polynomial is 3. This polynomial has 2 terms, so it is a binomial. This polynomial has 3 terms, so it is a trinomial.

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Writing Polynomials in Order The terms of a polynomial are usually arranged so that the powers of one variable are in ascending (increasing) order or descending (decreasing) order.

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Ex. 3 – Arrange the terms of the polynomial so that the powers of x are in descending order. 6x – 7x 5 + 4x 2 + 9 Find the degree of each term. Then arrange them in decreasing order: 6x – 7x 5 + 4x 2 + 9 –7x 5 + 4x 2 + 6x + 9 Degree1 52 0 5 2 1 0 The polynomial written in descending order is -7x 5 + 4x 2 + 6x + 9.

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Ex. 4 - Write the terms of the polynomial so that the powers of x are in descending order. Find the degree of each term. Then arrange them in decreasing order: y 2 + y 6 − 3y y 2 + y 6 – 3y y 6 + y 2 – 3y Degree 2 6 1 6 2 1 The polynomial written in descending order is y 6 + y 2 – 3y.

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6-2 Adding and Subtracting Polynomials Algebra 1 ~ Chapter 8.5 “Adding and Subtracting Polynomials”

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6-2 Adding and Subtracting Polynomials Warm Up - Simplify each expression by combining like terms. 1. 4x + 2x 2. 3y + 7y 3. 8p – 5p 4. 5n + 6n 2 5. 3x 2 + 6x 2 6. 12xy – 4xy 6x6x 10y 3p3p Not like terms 9x29x2 8xy

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o Just as you can perform operations on numbers, you can perform operations on polynomials. o To add or subtract polynomials, combine like terms.

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Example 1: Adding and Subtracting Monomials A. 12p 3 + 11p 2 + 8p 3 12p 3 + 8p 3 + 11p 2 20p 3 + 11p 2 B. 5x 2 – 6 – 3x + 8 5x 2 – 3x + 8 – 6 5x 2 – 3x + 2 Arrange the terms so the “like” terms are next to each other and then simplify.

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Polynomials can be added in either vertical or horizontal form. Simplify (5x 2 + 4x + 1) + (2x 2 + 5x + 2) In vertical form, align the like terms and add: 5x 2 + 4x + 1 + 2x 2 + 5x + 2 7x 2 + 9x + 3

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In horizontal form, regroup and combine like terms. (5x 2 + 4x + 1) + (2x 2 + 5x + 2) = (5x 2 + 2x 2 ) + (4x + 5x) + (1 + 2) = 7x 2 + 9x + 3

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Example 2: Adding Polynomials A. (4m 2 + 5m + 1) + (m 2 + 3m + 6) (4m 2 + 5m + 1) + (m 2 + 3m + 6) (4m 2 + m 2 ) + (5m + 3m) + (1 + 6) 5m 2 + 8m + 7 B. (10xy + x) + (–3xy + y) (10xy + x) + (–3xy + y) (10xy – 3xy) + x + y 7xy + x + y

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Example 2: Adding Polynomials C.

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Subtracting Polynomials Simplify (4x + 5) – ( 2x + 1) (4x – 2x) + (5 – 1 ) 2x + 4 (4x + 5) + (-2x – 1) (4x + -2x) + (5 + -1) 2x + 4 Option #1:Option #2: Recall that you can subtract a number by adding its opposite.

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Example 3: Subtracting Polynomials A. (4m 2 + 5m + 1) − (m 2 + 3m + 6) (4m 2 + 5m + 1) − (m 2 + 3m + 6) (4m 2 − m 2 ) + (5m − 3m) + (1 − 6) 3m 2 + 2m – 5 B. (10x 3 + 5x + 6) − (–3x 3 + 4) (10x 3 - - 3x 3 ) + (5x – 0x) + (6 – 4) 13x 3 + 5x + 2

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Example 3C: Subtracting Polynomials (7m 4 – 2m 2 ) – (5m 4 – 5m 2 + 8) (7m 4 – 5m 4 ) + (−2m 2 – − 5m 2 ) + (0 – 8) (7m 4 – 5m 4 ) + (–2m 2 + 5m 2 ) – 8 2m 4 + 3m 2 – 8

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Example 3D: Subtracting Polynomials (–10x 2 – 3x + 7) – (x 2 – 9) (–10x 2 – x 2 ) + (−3x – 0x) + (7 – -9) –11x 2 – 3x + 16

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Lesson Wrap Up Simplify each expression. 1. 7m 2 + 3m + 4m 2 2. (r 2 + s 2 ) – (5r 2 + 4s 2 ) 3. (10pq + 3p) + (2pq – 5p + 6pq) 4. (14d 2 – 8) – (6d 2 – 2d + 1) –4r 2 – 3s 2 11m 2 + 3m 18pq – 2p 8d 2 +2d – 9 5. (2.5ab + 14b) – (–1.5ab + 4b)4ab + 10b

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Assignment Study Guide 8-4 (In-Class) Study Guide 8-5 (In-Class) Skills Practice 8-4 (Homework) Skills Practice 8-5 (Homework)

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