1 / 57

Polynomials

Polynomials. Algebra I. Vocabulary. Monomial – a number, variable or a product of a number and one or more variables. Binomial – sum of two monomials. Trinomial – sum of three monomials. Polynomial – a monomial or a sum of monomials. Constants – monomials that are real numbers.

ivo
Download Presentation

Polynomials

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Polynomials Algebra I

  2. Vocabulary • Monomial – a number, variable or a product of a number and one or more variables. • Binomial – sum of two monomials. • Trinomial – sum of three monomials. • Polynomial – a monomial or a sum of monomials. • Constants – monomials that are real numbers.

  3. Examples • Monomial • 13n • -5z • Binomial • 2a + 3c • 6x²+ 3xy • Trinomial • p²+ 5p + 4 • 3a²- 2ab - b²

  4. More Vocabulary • Terms – the individual monomial. • Degree of a monomial – sum of the exponents of all it’s variables. • Degree of a polynomial – The greatest degree of any term in the polynomial. To find the degree of a polynomial, you must first find the degree of each term.

  5. Examples • Monomial • 8y³ degree is 3 • 3a degree is 1 • Polynomial • 5mn³degree is 4 • -4x²y²+ 3x²y + 5 degree is 4 • 3a + 7ab – 2a²b + 16 degree is 3

  6. Ordering Polynomials • Ascending Order – ordering polynomials based on their exponents least to greatest. You will only look at one variable’s exponents, usually ‘x’, unless told otherwise. • Descending Order – ordering polynomials based on their exponents greatest to least.

  7. Ascending Example • Arrange polynomials in ascending order 7x³+ 2x²- 11 Look at the exponents of the variable, choose the lowest. When there isn’t a variable with a term, it’s like there is an exponent of zero. 7x³+ 2x²- 11x° -11 + 2x²+ 7x³

  8. Descending Example • Arrange polynomials in descending order 6x²+ 5 – 8x – 2x³ 6x²+ 5x°- 8x – 2x³ -2x³+ 6x²- 8x + 5

  9. Now you try… Arrange in descending order 9 + 4x³- 3x – 10x² Arrange in ascending order 10y³- 4y + 16

  10. Now you try… Arrange in descending order 9 + 4x³- 3x – 10x² 4x³- 10x²- 3x + 9 Arrange in ascending order 10y³- 4y + 16 16 – 4y + 10y³

  11. Adding Polynomials • Group like terms and combine (add) the coefficients. (3x²- 4x + 8) + (2x – 7x²- 5)

  12. Adding Polynomials • Group like terms and combine (add) the coefficients. (3x²- 4x + 8) + (2x – 7x²- 5) 3x²- 4x + 8+ 2x – 7x²- 5 Find like terms -4x²- 2x + 3 Combine like terms (add)

  13. Adding Polynomials • Group like terms and combine (add) the coefficients. (3x²- 4x + 8) + (2x – 7x²- 5) 3x²- 4x + 8+ 2x – 7x²- 5 Find like terms -4x²-2x + 3 Combine like terms (add) • Final answers must be in descending order

  14. Now You Try… (-2x³+ 3x²- 15x + 3) + (7x²+ 9x – 10) (4 – 6x²+ 12x) + (8x²- 3x – 5)

  15. Now You Try… (-2x³+ 3x²- 15x + 3) + (7x²+ 9x – 10) -2x³+ 10x²- 6x – 7 (4 – 6x²+ 12x) + (8x²- 3x – 5) 2x²+ 9x -1

  16. Subtracting Polynomials • Distribute the negative (minus) to the second set of parenthesis (include EVERYTHING in the parenthesis). (3n²+ 13n³+ 5n) – (7n – 4n³)

  17. Subtracting Polynomials • Distribute the negative (minus) to the second set of parenthesis. (3n²+ 13n³+ 5n) – (7n – 4n³) • This will change the signs of everything in the second set of parenthesis.

  18. Subtracting Polynomials • Distribute the negative (minus) to the second set of parenthesis. (3n²+ 13n³+ 5n) – (7n – 4n³) 3n²+ 13n³+ 5n – 7n + 4n³ • Next combine like terms to simplify.

  19. Subtracting Polynomials (3n²+ 13n³+ 5n) – (7n – 4n³) 3n²+ 13n³+ 5n – 7n + 4n³ 3n²+13n³+ 5n – 7n + 4n³ 17n³+ 3n²- 2n • The answer should be in descending order

  20. Now You Try… (4x²+ 8x – 2) – (-5x – 2x²+ 7) (10a³- 2a²+ 12a) – (7a²+ 3a – 10)

  21. Now You Try… (4x²+ 8x – 2) – (-5x – 2x²+ 7) 4x²+ 8x - 2 + 5x + 2x²- 7 6x²+ 13x – 9 (10a³- 2a²+ 12a) – (7a²+ 3a – 10) 10a³- 2a²+ 12a – 7a²- 3a + 10 10a³- 9a²+ 9a + 10

  22. Review Problems Before the Quiz (-2x²- 4x + 7) + (8x²+ 10x – 8) (5x – 3) – (3x²+ 5x – 10)

  23. Review Problems Before the Quiz (-2x²- 4x + 7) + (8x²+ 10x – 8) -2x²- 4x + 7 + 8x²+ 10x – 8 6x²+ 6x - 1 (5x – 3) – (3x²+ 5x – 10) 5x – 3 – 3x²- 5x + 10 -3x²+ 7

  24. Multiplying a Polynomial by a Monomial • Use the distributive property. • Multiply the monomial by EVERYTHING in the parenthesis. • Don’t forget your rules for multiplying like bases and exponents! You will add the exponents. • Combine like terms when necessary

  25. Multiplying a Polynomial by a Monomial -2x²(3x²- 7x + 10) -6x + 14x³- 20x² • The answer should be in descending order.

  26. Now You Try… 2x(-6x²+ 7x – 10) (4x²- 8x + 3)(-5x)

  27. Now You Try… 2x(-6x²+ 7x – 10) -12x³+ 14x²- 20x (4x²- 8x + 3)(-5x) -20x³+ 40x² - 15x

  28. Using Multiple Operations in Polynomials • Always follow PEMDAS 7x(3x²- 5x + 7) + 4x²(3x – 1)

  29. Using Multiple Operations in Polynomials • Always follow PEMDAS • Multiply before adding/subtracting • Use distributive property 7x(3x²- 5x + 7) + 4x²(3x – 1) 21x³- 35x²+ 49x + 12x³- 4x²

  30. Using Multiple Operations in Polynomials • Always follow PEMDAS • Multiply before adding/subtracting • Use the distributive property • Next, combine like terms 7x(3x²- 5x + 7) + 4x²(3x – 1) 21x³- 35x²+ 49x + 12x³- 4x² 33x³- 39x²+ 49x • Answers should be in descending order

  31. Now You Try… -2x(10x²- 7x + 4) + 3(-2x²+ 6x) 4x²(3x – 15) – 3x(11x³+ 5x²- 10)

  32. Now You Try… -2x(10x²- 7x + 4) + 3(-2x²+ 6x) -20x³+ 14x²- 8x – 6x²+ 18x 4x²(3x – 15) – 3x(11x³+ 5x²- 10) 12x³- 60x²- 33x – 15x³+ 30x

  33. Now You Try… -2x(10x²- 7x + 4) + 3(-2x²+ 6x) -20x³+ 14x²- 8x – 6x²+ 18x -20x³+ 8x²+ 10x 4x²(3x – 15) – 3x(11x³+ 5x²- 10) 12x³- 60x²- 33x – 15x³+ 30x -33x – 3x³- 60x²+ 30x

  34. Multiplying Polynomials Using FOIL • When multiplying a binomial with a binomial you can use the FOIL method to simplify. First Outer Inner Last • This is a way to remember to multiply each term of the expression.

  35. Multiplying Polynomials Using FOIL • Multiply the First terms in each binomial (1x + 1)(1x + 2) 1x² • Use the rules for monomials when multiplying! • Don’t forget to put a “1” in before the variable, if there is not a coefficient.

  36. Multiplying Polynomials Using FOIL • Multiply the Outer terms in each binomial (1x + 1)(1x + 2) 1x²+ 2x • Use the rules for monomials when multiplying!

  37. Multiplying Polynomials Using FOIL • Multiply the Inner terms in each binomial (1x + 1)(1x + 2) 1x²+ 2x + 1x • Use the rules for monomials when multiplying!

  38. Multiplying Polynomials Using FOIL • Multiply the Last terms in each binomial (1x + 1)(1x + 2) 1x²+ 2x + 1x + 2 • Use the rules for monomials when multiplying!

  39. Multiplying Polynomials Using FOIL • Now combine like terms. (1x + 1)(1x + 2) 1x²+ 2x + 1x + 2 • Always put your answer in descending order!

  40. Multiplying Polynomials Using FOIL • Now combine like terms. (1x + 1)(1x + 2) 1x²+ 2x + 1x + 2 1x²+ 3x + 2 • Always put your answer in descending order!

  41. Now You Try… (2x – 5)(x + 4) (-4x – 8)(3x – 2)

  42. Now You Try… (2x – 5)(x + 4) 2x²+ 8x – 5x - 20 2x² + 3x - 20 (-4x – 8)(3x – 2) -12x²+ 8x – 24x + 16 -12x²– 16x + 16

  43. Special Products • Square of a sum or difference (4y + 5)² • The ENTIRE polynomial has to be squared. (4y + 5)(4y + 5) • Then use FOIL to solve.

  44. Special Products • Square of a sum or difference (4y + 5)² (4y + 5)(4y + 5) 16y²+ 20y + 20y + 25 • Combine like terms

  45. Special Products • Square of a sum or difference (4y + 5)² (4y + 5)(4y + 5) 16y²+ 20y + 20y + 25 16y²+ 40y + 25 • Final answer should be in descending order.

  46. Special Products • Product of a sum and a difference • The binomials are the same except one is plus and one is minus. (3n + 2)(3n – 2) • Use FOIL to simplify

  47. Special Products • Product of a sum and a difference (3n + 2)(3n – 2) 9n²- 6n + 6n - 4 9n²- 4 • Combine like terms • Notice that the two center terms (like terms) will cancel each other out.

  48. Now You Try… (8c + 3d)² (5m³- 2n)² (11v – 8w)(11v + 8w)

  49. Now You Try… (8c + 3d)² (8c + 3d)(8c + 3d) 64c²+ 24cd + 24cd + 9d² 64c²+ 48cd + 9d² (5m³- 2n)² (5m³- 2n)(3m³- 2n) 15m – 10mn – 10mn + 4n² 15m – 20mn + 4n²

  50. Now You Try… (11v – 8w)(11v + 8w) 11v²+ 88vw – 88vw – 64w² 11v²- 64w²

More Related