1. Polynomial Basics Adding and Subtracting Polynomials MM1A2 a

3. Collect or combine the like terms: 3x – 6 + 2x – 8 1) 4x 2 + 3x – 7 - 4x 2 + 2x 2) 5x 2 y + 10xy - 2x 2 y + 5y – 6xy 5x - 14 5x - 7 3x 2 y + 4xy +5y

4. The first thing we will do with polynomials is classify. There are two ways we can do this: – By the number of terms – By the degree Terms are the “chunks” of numbers and/or variables separated by the + or – signs. The degree of a polynomial is the largest exponent on a variable.

5. What is a TERM?? The basic unit of algebra is a term. 6a-xy4b 2 -0.7p A term has three (3) parts: 1.A sign, being positive(+) or negative (-). 2.A number, called the coefficient. 3.It may or may not contain a letter, called the variable.

6. What is a TERM?? Every term must have a sign. If there isn’t one, assume there is an unwritten positive sign. Negative signs will always be written. 2x-4s3x-9xy + 2x + 3x

7. What is a TERM? Every term MUST have a number. If there isn’t one, assume the unwritten coefficient of one (1). What is the coefficient of each term? -x2y-4xyx 2 2 -41

8. What is a TERM?? Every term may or may NOT have a variable. If there is a variable and it has an exponent above the variable, 2x 2, then the exponent belongs ONLY to the variable and not the whole term. EXAMPLES 3x-x 2 (4x) 5 3xy 2

9. Complete the chart below: Example Sign (+ or -) CoefficientVariable(s) 8f -10r 4xy -22 -xyz

10. Classifying by terms… # of termsClassifying Word Example 1Monomial 2*Binomial 3*Trinomial 4 or more*Polynomial* I suppose mathematicians lost interest in naming polynomials after 3 terms. Anything 4 terms or more gets the boring label of plain ol’ polynomial. Remember, terms are separated by a + or - sign, which then becomes a positive or negative for the term to follow!!

11. What is NOT a polynomial term???

12. Polynomials A polynomial is an expression which consists of one or more terms. So… – A monomial is a polynomial – A binomial is a polynomial – A trinomial is a polynomial

13. Classify each of the following according to the number of terms. 1.3xy 2.2x + 3y - 6 3.x 2 - 3y + 9 - 6x 4.4x 2 y 2 z 4 5.2x -5 6.9 monomial trinomial polynomial monomial binomial Constant; NOT a polynomial

14. Classifying by degree To classify by degree you must know the difference between a single variable term, such as 3x and a multi-variable term such as 3x 2 y 2. Example: 2z 2 3xy 3 -x 2 y 3 z 4 5m 6 Single variable term Multi variable term Single variable term

15. single variable terms To determine the degree of a single variable polynomial, simply look for the term with the largest exponent. Example: 4x + 3x 3 + 9y + 4 3rd degree 7y 2 + 3x - 92nd degree 8x 4 + 9y 9 9th degree

16. multi-variable terms To determine the degree of a multi-variable polynomial, you must add all the exponents within a each individual term, then take the highest number. Example: 3x 2 y 4 + 5xy 8 + 5 9th degree 5xy 2 z 5 + 3x 3 y 4 8th degree 4 + 2 = 61 + 8 = 9 1 + 2 + 5 = 83 + 4 = 7

17. C

18. Try these: 1.2x 2 + 3x - 9 2.4x 2 y 4 + 2xy 2 + 9 3.5x - 16 4.2x 2 y 2 - 9x + 2y 7 5.7 6.X 2 + 2x 2 y + 4y 2 Degree = 2 Degree = 6 Degree = 1 Degree = 7 Degree = 0 Degree = 3

19. Classifying by degree… Degree (largest exponent on a variable) Classifying WordExample 0Constant 9 (yep, a plain ol’ number) 1Linear 2Quadratic 3Cubic 4Quartic Polynomials with a degree higher than 4 are not named at this level !!

20. Your Turn… Classify by Term & Degree 4 Terms – Polynomial Degree of 3 - Cubic

21. Your Turn… Classify by Term & Degree 2 Terms – Binomial Degree of 1 - Linear

22. Your Turn… Classify by Term & Degree 3 Terms – Trinomial Degree of 2 - Quadratic

23. Your Turn… Classify by Term & Degree 2 Terms – Binomial Degree of 4 - Quartic

24. Your Turn… Classify by Term & Degree 1 Term – Monomial Degree of 0 - Constant Notice that classifying by terms has NOTHING to do with classifying by degree!

25. Does order matter? Polynomials are usually arranged in one of two ways: – Ascending order (smallest degree to largest degree) – Descending order (largest degree to smallest degree) When a polynomial is written in descending order the coefficient of the first term is called the leading coefficient.

26. Essential Question…

27. Operations with Polynomials Adding polynomials is simply combining like terms. (Like terms have the same exact variable and degree of exponents!) Example: 1) List terms in descending order 3x 4 - 2x 3 + 9x 3 + 5x 2 + 7x 2 - 2x 2) Add or subtract the coefficients of like terms: 3x 4 + 7x 3 + 12x 2 - 2x

29. 4

30. More Practice Match the polynomial to the correct degree!!

32. 1. 2n 2 + 6n - 8 D.

33. 2. D

34. 3. (2r 3 + 12r 2 - r) + (3r 3 + 7r - 6) A

35. 4. A

36. 5. C

37. 6. A

38. 7. B

39. Essential Question…

40. Example Tip: Be careful with subtraction…watch your signs! 11x 2 - 2x 2 + 6x + 7x - 5 9x 2 + 13x = 5

45. Complete Adding and Subtracting Polynomials Worksheet!! Do your best on this worksheet and bring them back tomorrow!!

46. Multiplying Polynomials!!

47. Operations with Polynomials Multiplication is basically distribution.

48. Another Example