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9.0 Classifying Polynomials

9.0 Classifying Polynomials . Objective: SWBAT describe and write polynomials in standard form. Concept: Unit 9 – Polynomials and Factoring . MONOMIAL. A number, a variable, or the product/quotient of numbers and variables 4  is a monomial 3x  is a monomial 14x 6  is a monomial.

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9.0 Classifying Polynomials

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  1. 9.0 Classifying Polynomials • Objective: SWBAT describe and write polynomials in standard form. • Concept: Unit 9 – Polynomials and Factoring

  2. MONOMIAL A number, a variable, or the product/quotient of numbers and variables • 4  is a monomial • 3x  is a monomial • 14x6  is a monomial

  3. Degree of a Monomial The exponent of the variable.

  4. Degree of Monomial 0 1 2 5 Monomial 3 4x -5x2 18x5

  5. Degree of aMonomialwith more than one variable… The sum of the exponents of all of its variables. 7xyz = degree 3

  6. Polynomial A monomial or the sum/difference of monomials which contain only 1 variable. The variable cannot be in the denominator of a term.

  7. Degree of a Polynomial The degree of the term with the highest degree; what is the highest exponent??

  8. Degree of Polynomial 2 4 1 3 Polynomial 6x2 - 3x + 2 15 - 4x + 5x4 x + 10 + x x3 + x2 + x + 1

  9. Standard Form of a Polynomial A polynomial written so that the degree of the terms decreases from left to right and no terms have the same degree.

  10. Not Standard 6x + 3x2 - 2 15 - 4x + 5x4 x + 10 + x 1 + x2 + x + x3 Standard 3x2 + 6x - 2 5x4 - 4x + 15 2x + 10 x3 + x2 + x + 1

  11. Naming Polynomials Polynomials are named or classified by their degree and the number of terms they have.

  12. Degree Name constant linear quadratic cubic Polynomial 7 5x + 2 4x2 + 3x - 4 6x3 - 18 Degree 0 1 2 3 For degrees higher than 3 say: “nth degree” x5 + 3x2 + 3 “5th degree” x8 + 4 “8th degree”

  13. # Terms Name monomial binomial trinomial binomial Polynomial 7 5x + 2 4x2 + 3x - 4 6x3 - 18 # Terms 1 2 3 2 For more than 3 terms say: “a polynomial with n terms” or “an n-term polynomial” 11x8 + x5 + x4 - 3x3 + 5x2 - 3 “a polynomial with 6 terms” – or – “a 6-term polynomial”

  14. Polynomial -14x3 -1.2x2 -1 7x - 2 3x3+ 2x - 8 2x2 - 4x + 8 x4 + 3 Name cubic monomial quadratic monomial constant monomial linear binomial cubic trinomial quadratic trinomial 4th degree binomial

  15. 9.1 Adding and Subtracting Polynomials • Objective: SWBAT add and subtract complex polynomials. • Concept: Unit 9 – Polynomials and Factoring

  16. Adding Polynomials 9.1 Adding and Subtracting Polynomials in horizontal form: (x2 + 2x + 5) + (3x2 + x + 12) = 4x2 + 3x + 17 in vertical form: x2 + 2x + 5 + 3x2 + x + 12 = 4x2 + 3x + 17

  17. 9.1 Adding and Subtracting Polynomials 3x2 – 4x + 8 x2 + x + 1 + 2x2 – 7x – 5 Simplify: Then classify the polynomial by number of terms and degree + 2x2 + 3x + 2 5x2 – 11x + 3 3x2 + 4x + 3 (4b2 + 2b + 1) + (7b2 + b – 3) a2 + 8a – 5 + 3a2 + 2a – 7 11b2 + 3b – 2 4a2 + 10a – 12

  18. 9.1 Adding and Subtracting Polynomials 7d2 + 7d z2 + 5z + 4 + 2d2 + 3d Simplify: Then classify the polynomial by number of terms and degree + 2z2 - 5 9d2 + 10d 3z2 + 5z – 1 (2x2 – 3x + 5) + (4x2 + 7x – 2) (a2 + 6a – 4) + (8a2 – 8a) 9a2 – 2a – 4 6x2 + 4x + 3

  19. 9.1 Adding and Subtracting Polynomials 3d2 + 5d – 1 7p2 + 5 + (–4d2– 5d + 2) Simplify: Then classify the polynomial by number of terms and degree + (–5p2 – 2p + 3) –d2 + 1 2p2– 2p + 8 (3x2 + 5x) + (4 – 6x – 2x2) (x3 + x2 + 7) + (2x2 + 3x – 8) x2– x + 4 x3 + 3x2 + 3x – 1

  20. 9.1 Adding and Subtracting Polynomials 4p2 + 5p 2x3 + x2 – 4 Simplify: + (-2p + p + 7) + 3x2 – 9x + 7 4p2 + 4p + 7 2x3 + 4x2 – 9x + 3 5y2 – 3y + 8 + 4y3 – 9 (8cd – 3d + 4c) + (-6 + 2cd – 4d) 4y3 + 5y2 – 3y – 1 4c – 7d + 10cd – 6

  21. 9.1 Adding and Subtracting Polynomials Simplify: (12y3 + y2 – 8y + 3) + (6y3 – 13y + 5) (7y3 + 2y2 – 5y + 9) + (y3 – y2 + y – 6) (6x5 + 3x3 – 7x – 8) + (4x4 – 2x2 + 9) = 18y3 + y2 – 21y + 8 = 8y3 + y2 – 4y + 3 = 6x5 + 4x4 + 3x3 – 2x2 – 7x + 1

  22. Subtracting Polynomials ubtractionis adding the opposite) (5x2 + 10x + 2) – (x2 – 3x + 12) (5x2 + 10x + 2) + (–x2) + (3x) + (–12) = 4x2 + 13x – 10 5x2 + 10x + 2 = 5x2 + 10x + 2 –(x2 – 3x + 12) = –x2 + 3x – 12 4x2 + 13x – 10

  23. Simplify: 9.1 Adding and Subtracting Polynomials (3x2 – 2x + 8) – (x2 – 4) 3x2 – 2x + 8 + (–x2) + 4 = 2x2 – 2x + 12 (10z2 + 6z + 5) – (z2 – 8z + 7) 10z2 + 6z + 5 + (–z2) + 8z + (–7) = 9z2 + 14z – 2

  24. Simplify: 9.1 Adding and Subtracting Polynomials 3x2 – 7x + 5 7a2 – 2a – (x2 + 4x + 7) – (5a2 + 3a) 2x2 – 11x – 2 2a2 – 5a 7x2 – x + 3 4x2 + 3x + 2 – (3x2 – x – 7) – (2x2 – 3x – 7) 2x2 + 6x + 9 4x2 + 10

  25. Simplify: 9.1 Adding and Subtracting Polynomials 2x2 + 5x 3x2 – 2x + 10 – (x2 – 3) – (2x2 + 4x – 6) x2 – 6x + 16 x2 + 5x + 3 4x2 – x + 6 3x2 – 5x + 3 – (3x2 – 4) – (2x2 – x – 4) x2 – 4x + 7 x2 – x + 10

  26. Simplify: 9.1 Adding and Subtracting Polynomials (4x5 + 3x3 – 3x – 5) – (–2x3 + 3x2 + x + 5) = 4x5 + 5x3 – 3x2 – 4x – 10 (4d4 – 2d2 + 2d + 8) – (5d3 + 7d2 – 3d – 9) = 4d4 – 5d3 – 9d2 + 5d + 17 (a2 + ab– 3b2) – (b2 + 4a2 – ab) = –3a2 + 2ab – 4b2

  27. Simplify by finding the perimeter: 9.1 Adding and Subtracting Polynomials x2 + x c2 + 1 6c + 3 A B 2x2 2x2 4c2 + 2c + 5 x2 + x A = 5c2 + 8c + 9 B = 6x2 + 2x

  28. Simplify by finding the perimeter: 9.1 Adding and Subtracting Polynomials 2d2 + d – 4 3x2 – 5 x D C 3d2 – 5d x2 + 7 3d2 – 5d d2 + 7 D = 9d2 – 9d + 3 C = 8x2 – 10x + 14

  29. Simplify by finding the perimeter: 9.1 Adding and Subtracting Polynomials 3x2 – 5x a3 + 2a a + 1 E F x2 + 3 2a3 + a + 3 E = 3a3 + 4a + 4 F = 8x2 – 10x + 6

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