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Problems of the Day Simplify each expression below. 1. = y 13. = -10d 7. 2. = – 72a 33 b 14. 3. Problems of the Day Simplify each expression below. 4.) 5.) 6.). Algebra 1 ~ Chapter 8.4. Polynomials.

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  1. Problems of the DaySimplify each expression below. 1. = y13 = -10d7 2. = – 72a33b14 3.

  2. Problems of the DaySimplify each expression below. 4.)5.)6.)

  3. Algebra 1 ~ Chapter 8.4 Polynomials

  4. 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.

  5. A. 4p4q3 Ex. 1 - Find the degree of each monomial. Add the exponents of the variables: 4 + 3 = 7. The degree is 7. B. 7ed A variable written without an exponent has an exponent of 1. 1+ 1 = 2. The degree is 2. C. 3 There is no variable, but you can write 3 as 3x0. The degree is 0.

  6. * 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.

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

  8. Ex. 2 – Find the degree of each polynomials. Then name the polynomials based on # of terms. A.) 5m4 + 3m B.) -4x3y2 + 3x2 + 5 C.) 3a + 7ab – 2a2b This polynomial has 2 terms, so it is a binomial. The greatest degree is 4, so the degree of the polynomial is 4. This polynomial has 3 terms, so it is a trinomial. The degree of the polynomial is 5. The degree of the polynomial is 3. This polynomial has 3 terms, so it is a trinomial.

  9. 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.

  10. 6x – 7x5 + 4x2 + 9 –7x5 + 4x2 + 6x + 9 2 Degree 1 5 2 5 1 0 0 Ex. 3 – Arrange the terms of the polynomial so that the powers of x are in descending order. 6x – 7x5 + 4x2 + 9 Find the degree of each term. Then arrange them in decreasing order: The polynomial written in descending order is -7x5 + 4x2 + 6x + 9.

  11. y2 + y6 – 3y y6 + y2 – 3y Ex. 4 - Write the terms of the polynomial so that the powers of x are in descending order. y2 + y6 − 3y Find the degree of each term. Then arrange them in decreasing order: Degree 2 6 1 6 2 1 The polynomial written in descending order is y6 + y2 – 3y.

  12. Algebra 1 ~ Chapter 8.5 “Adding and Subtracting Polynomials”

  13. Warm Up - Simplify each expression by combining like terms. 1.4x + 2x 2. 3y + 7y 3. 8p – 5p 4. 5n + 6n2 5. 3x2 + 6x2 6. 12xy – 4xy 6x 10y 3p Not like terms 9x2 8xy

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

  15. Example 1: Adding and Subtracting Monomials A. 12p3 + 11p2 + 8p3 Arrange the terms so the “like” terms are next to each other and then simplify. 12p3 + 8p3 + 11p2 20p3 + 11p2 B. 5x2 – 6 – 3x + 8 5x2 – 3x+ 8 – 6 5x2 – 3x + 2

  16. 5x2+ 4x+ 1 + 2x2+ 5x+ 2 Polynomials can be added in either vertical or horizontal form. Simplify (5x2 + 4x + 1) + (2x2 + 5x + 2) In vertical form, align the like terms and add: 7x2+ 9x+ 3

  17. In horizontal form, regroup and combine like terms. (5x2 + 4x + 1) + (2x2 + 5x + 2) = (5x2 + 2x2) + (4x + 5x) + (1 + 2) = 7x2+ 9x+ 3

  18. Example 2: Adding Polynomials A. (4m2 + 5m + 1) + (m2 + 3m + 6) (4m2+ 5m + 1) + (m2+3m+ 6) (4m2+m2) + (5m + 3m)+ (1 + 6) 5m2 + 8m + 7 B. (10xy + x) + (–3xy + y) (10xy + x) + (–3xy + y) (10xy– 3xy) + x +y 7xy+ x +y

  19. Example 2: Adding Polynomials C.

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

  21. Example 3: Subtracting Polynomials A. (4m2 + 5m + 1) − (m2 + 3m + 6) (4m2+ 5m + 1) − (m2+3m+ 6) (4m2−m2) + (5m−3m)+ (1 − 6) 3m2 + 2m – 5 B. (10x3 + 5x + 6) − (–3x3 + 4) (10x3 - - 3x3) + (5x – 0x) + (6 – 4) 13x3 + 5x + 2

  22. Example 3C: Subtracting Polynomials (7m4 – 2m2) – (5m4 – 5m2 + 8) (7m4 – 5m4)+ (−2m2 – −5m2) +(0 – 8) (7m4 – 5m4) + (–2m2 + 5m2)– 8 2m4 + 3m2 – 8

  23. Example 3D: Subtracting Polynomials (–10x2 – 3x + 7) – (x2 – 9) (–10x2 – x2) + (−3x – 0x) + (7 – -9) –11x2 – 3x + 16

  24. Lesson Wrap Up Simplify each expression. 1. 7m2 + 3m + 4m2 2. (r2 + s2) – (5r2 + 4s2) 3. (10pq + 3p) + (2pq – 5p + 6pq) 4. (14d2 – 8) – (6d2 – 2d + 1) 11m2 + 3m –4r2 – 3s2 18pq – 2p 8d2 +2d – 9 5. (2.5ab + 14b) – (–1.5ab + 4b) 4ab + 10b

  25. 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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