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  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