1. Splash Screen

2. Lesson Menu Five-Minute Check (over Chapter 7) CCSS Then/Now New Vocabulary Example 1:Identify Polynomials Example 2:Standard Form of a Polynomial Example 3:Add Polynomials Example 4:Subtract Polynomials Example 5:Real-World Example: Add and Subtract Polynomials

3. CCSS Content Standards A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients. A.APR.1 Understand 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. Mathematical Practices 3 Construct viable arguments and critique the reasoning of others. Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

4. Then/Now You identified monomials and their characteristics. Write polynomials in standard form. Add and subtract polynomials.

5. Vocab polynomial binomial trinomial degree of a monomial degree of a polynomial standard form of a polynomial leading coefficient

6. Term - a monomial (a number, a variable or a product of a number and variable) Polynomial is a sum of terms of the form axⁿ where n is a nonnegative integer.

7. # of Terms 1 term monomial 2 terms binomial 3 terms trinomial 4 or more terms polynomial with _ terms

8. Example 1 Identify Polynomials State whether each expression is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

9. Polynomial in standard form 2x³ + 5x² - 4x + 7 Degree of each term – sum of the exponents of the variables. Degree of 2x³y²z is (3+2+1)=6 Degree of polynomial – largest degree of any of its terms. Coefficient is the numerical factor of a term. Constant – If a term contains only a number like 7 it is called a constant Standard form-terms in order from greatest to least degree Leading Coefficient Degree Constant Term

10. Term vs. Coefficient TermCoefficientDegree -12x³-123 x³y14 -z-11 220

11. There are two ways to classify polynomials: 1 st by degree – find the largest degree 2 nd by the number of terms – count the number of terms Polynomial Classifications

12. By Degree Constant (0) Linear (1) Quadratic (2) Cubic (3) Quartic (4) Quintic (5) Polynomial ___degree (6 or more) By Degree

13. Anatomy of a Polynomial Is it in standard form? What is the leading coefficient? Classify by terms Classify by Degree What is the constant term?

14. Anatomy of a Polynomial Is it in standard form? What is the leading coefficient? Classify by terms Classify by Degree What is the constant term?

15. Anatomy of a Polynomial Is it in standard form? What is the leading coefficient? Classify by terms Classify by Degree What is the constant term?

16. Anatomy of a Polynomial Is it in standard form? What is the leading coefficient? Classify by terms Classify by Degree What is the constant term?

17. Anatomy of a Polynomial Is it in standard form? What is the leading coefficient? Classify by terms Classify by Degree What is the constant term?

18. Anatomy of a Polynomial Is it in standard form? What is the leading coefficient? Classify by terms Classify by Degree What is the constant term?

19. Anatomy of a Polynomial Is it in standard form? What is the leading coefficient? Classify by terms Classify by Degree What is the constant term?

20. Page 468 1-10,20-33,45-50 ASSIGNMENT

21. Example 1 A.yes, monomial B.yes, binomial C.yes, trinomial D.not a polynomial A. State whether 3x 2 + 2y + z is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

22. Example 1 A.yes, monomial B.yes, binomial C.yes, trinomial D.not a polynomial B. State whether 4a 2 – b –2 is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

23. Example 1 A.yes, monomial B.yes, binomial C.yes, trinomial D.not a polynomial C. State whether 8r – 5s is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

24. Example 1 A.yes, monomial B.yes, binomial C.yes, trinomial D.not a polynomial D. State whether 3y 5 is a polynomial. If it is a polynomial, identify it as a monomial, binomial, or trinomial.

25. Example 2 Standard Form of a Polynomial A. Write 9x 2 + 3x 6 – 4x in standard form. Identify the leading coefficient. Answer: 3x 6 + 9x 2 – 4x; the leading coefficient is 3. Step 2Write the terms in descending order. Step 1Find the degree of each term. Degree: 261 Polynomial:9x 2 + 3x 6 – 4x

26. Example 2 Standard Form of a Polynomial B. Write 12 + 5y + 6xy + 8xy 2 in standard form. Identify the leading coefficient. Answer: 8xy 2 + 6xy + 5y + 12; the leading coefficient is 8. Step 2Write the terms in descending order. Step 1Find the degree of each term. Degree: 0123 Polynomial: 12 + 5y + 6xy + 8xy 2

27. Example 2 A.3x 7 + 9x 4 – 4x 2 – 34x B. 9x 4 + 3x 7 – 4x 2 – 34x C. –4x 2 + 9x 4 + 3x 7 – 34x D.3x 7 – 4x 2 + 9x 4 – 34x A. Write –34x + 9x 4 + 3x 7 – 4x 2 in standard form.

28. Example 2 A.–72 B. 8 C. –6 D.72 B. Identify the leading coefficient of 5m + 21 –6mn + 8mn 3 – 72n 3 when it is written in standard form.

32. Adding & Subtracting Polynomials

33. Like Terms: terms that have identical variables both in letters and degree Combining Like Terms: adding and subtracting like terms – changes only the coefficient Adding and Subtracting Polynomials: simply combining like terms 2 ways to add and subtract 1.Horizontal 2.Vertical

34. Example 3 Add Polynomials A. Find (7y 2 + 2y – 3) + (2 – 4y + 5y 2 ). Horizontal Method (7y 2 + 2y – 3) + (2 – 4y + 5y 2 ) = (7y 2 + 5y 2 ) + [2y + (–4y)] + [(–3) + 2]Group like terms. = 12y 2 – 2y – 1Combine like terms.

35. Example 3 Add Polynomials Vertical Method Answer: 12y 2 – 2y – 1 7y 2 + 2y – 3 (+) 5y 2 – 4y + 2 Notice that terms are in descending order with like terms aligned. 12y 2 – 2y – 1

36. Example 3 Add Polynomials B. Find (4x 2 – 2x + 7) + (3x – 7x 2 – 9). Horizontal Method (4x 2 – 2x + 7) + (3x – 7x 2 – 9) = [4x 2 + (–7x 2 )] + [(–2x) + 3x] + [7 + (–9)]Group like terms. = –3x 2 + x – 2Combine like terms.

37. Example 3 Add Polynomials Vertical Method Answer: –3x 2 + x – 2 Align and combine like terms. 4x 2 – 2x + 7 (+) –7x 2 + 3x – 9 –3x 2 + x – 2

38. Adding & Subtracting (6x 2 – x + 3) + (-2x + x 2 – 7)

39. Adding & Subtracting (-8x 3 + x – 9x 2 + 2) + (8x 2 – 2x + 4) + (4x 2 – 1 – 3x 3 )

40. Example 4 Subtract Polynomials A. Find (6y 2 + 8y 4 – 5y) – (9y 4 – 7y + 2y 2 ). Horizontal Method Subtract 9y 4 – 7y + 2y 2 by adding its additive inverse. (6y 2 + 8y 4 – 5y) – (9y 4 – 7y + 2y 2 ) = (6y 2 + 8y 4 – 5y) + (–9y 4 + 7y – 2y 2 ) = [8y 4 + (–9y 4 )] + [6y 2 + (–2y 2 )] + (–5y + 7y) = –y 4 + 4y 2 + 2y

41. Example 4 Subtract Polynomials Vertical Method Align like terms in columns and subtract by adding the additive inverse. Answer: –y 4 + 4y 2 + 2y 8y 4 + 6y 2 – 5y (–)9y 4 + 2y 2 – 7y Add the opposite. 8y 4 + 6y 2 – 5y (+) –9y 4 – 2y 2 + 7y –y 4 + 4y 2 + 2y

42. Example 4 Subtract Polynomials Find (6n 2 + 11n 3 + 2n) – (4n – 3 + 5n 2 ). Answer: 11n 3 + n 2 – 2n + 3 Horizontal Method Subtract 4n 4 – 3 + 5n 2 by adding the additive inverse. (6n 2 + 11n 3 + 2n) – (4n – 3 + 5n 2 ) = (6n 2 + 11n 3 + 2n) + (–4n + 3 – 5n 2 ) = 11n 3 + [6n 2 + (–5n 2 )] + [2n + (–4n)] + 3 = 11n 3 + n 2 – 2n + 3

43. Example 4 Subtract Polynomials Vertical Method Align like terms in columns and subtract by adding the additive inverse. Answer: 11n 3 + n 2 – 2n + 3 11n 3 + 6n 2 + 2n + 0 (–) 0n 3 + 5n 2 + 4n – 3 Add the opposite. 11n 3 + 6n 2 + 2n + 0 (+) 0n 3 – 5n 2 – 4n + 3 11n 3 + n 2 – 2n + 3

44. Adding & Subtracting (6x 2 – x + 3) – (-2x + x 2 – 7)

45. Adding & Subtracting (-6x 3 + 5x – 3) – (2x 3 + 4x 2 – 3x + 1)

46. Adding & Subtracting (4x 2 – 1) – (3x – 2x 2 )

47. Adding & Subtracting (-8x 3 + x – 9x 2 + 2) + (8x 2 – 2x + 4) – (4x 2 – 1 – 3x 3 )

48. Page 469 #34-43,54,56 ASSIGNMENT

49. Example 3 A.–2x 2 + 5x + 3 B.8x 2 + 6x – 4 C.2x 2 + 5x + 4 D.–15x 2 + 6x – 4 A. Find (3x 2 + 2x – 1) + (–5x 2 + 3x + 4).

50. Example 3 A.5x 2 + 3x – 6 B.4x 3 + 5x 2 + 3x – 6 C.7x 3 + 5x 2 + 3x – 6 D. 7x 3 + 6x 2 + 3x – 6 B. Find (4x 3 + 2x 2 – x + 2) + (3x 2 + 4x – 8).

51. Example 4 A.2x 2 + 7x 3 – 3x 4 B.x 4 – 2x 3 + x 2 C.x 2 + 8x 3 – 3x 4 D.3x 4 + 2x 3 + x 2 A. Find (3x 3 + 2x 2 – x 4 ) – (x 2 + 5x 3 – 2x 4 ).

52. Example 4 A.2y 4 – 2y 2 – 11 B.2y 4 + 5y 3 + 3y 2 – 11 C.2y 4 – 5y 3 + 3y 2 – 11 D.2y 4 – 5y 3 + 3y 2 + 7 B. Find (8y 4 + 3y 2 – 2) – (6y 4 + 5y 3 + 9).

53. End of the Lesson