1. Polynomials 02/11/12lntaylor ©

2. Table of Contents Learning Objectives Adding Polynomials Distributing Negative Signs Multiplying Polynomials Special Case Binomials Shortcuts to Multiplying Binomials Dividing Polynomials Factoring Polynomials 02/11/12lntaylor ©

3. Adding Polynomials 02/11/12lntaylor © TOC

4. + x 2 3x 2 Step 1 Step 2 4x 2 + 3x+ 3 – 2x– 2 + x+ 1 Look for the same variable and exponent combinations Combine like terms in columns Step 3Add terms Note:When you add or subtract polynomials exponents do not change 02/11/12lntaylor © TOC

5. Now you try 10x 2 – 7x + 18 – 3x 2 – 3x – 7 02/11/12lntaylor © TOC

6. – 3x 2 10x 2 Step 1 Step 2 7x 2 – 7x+ 18 – 3x– 7 –10 x+ 11 Look for the same variable and exponent combinations Combine like terms in columns Step 3Add terms Note:When you add or subtract polynomials exponents do not change 02/11/12lntaylor © TOC

7. Now you try – x 2 – 7x – 18 – 8x 2 – 3x – 9 02/11/12lntaylor © TOC

8. – 8x 2 – x 2 Step 1 Step 2 – 9x 2 – 7x– 18 – 3x– 9 –10 x– 27 Look for the same variable and exponent combinations Combine like terms in columns Step 3Add terms Note:When you add or subtract polynomials exponents do not change 02/11/12lntaylor © TOC

9. Distributing Negative Signs 02/11/12lntaylor © TOC

10. Distributing a Red Flag 10x 2 – 7x + 18 – (3x 2 – 3x – 7) 02/11/12lntaylor © TOC

11. 10x 2 Step 4 Step 5 7x 2 – 7x+ 18 – 4x+ 25 Look for the same variable and exponent combinations Combine like terms in columns Step 6Add terms Note:When you add or subtract polynomials exponents do not change Step 2 Step 1 – ( ) means a red flag – mistake zone Add the – to each sign in the ( ) – + 3x 2 – – 3x– – 7 Step 3 Rewrite with one sign for each term – 3x 2 + 3x+ 7 – (3x 2 – 3x – 7) 02/11/12lntaylor © TOC

12. Now you try – 20x 2 + 10x – 18 – (– 5x 2 + 3x – 7) 02/11/12lntaylor © TOC

13. – 20x 2 Step 4 Step 5 – 15x 2 + 10x– 18 + 7x – 11 Look for the same variable and exponent combinations Combine like terms in columns Step 6Add terms Note:When you add or subtract polynomials exponents do not change Step 2 Step 1 – ( ) means a red flag – mistake zone Add the – to each sign in the ( ) – – 5x 2 – + 3x– – 7 Step 3 Rewrite with one sign for each term + 5x 2 – 3x + 7 – (– 5x 2 + 3x – 7) 02/11/12lntaylor © TOC

14. Now you try 12x 2 + 14x – 10 – (4x 2 – 2x – 7) 02/11/12lntaylor © TOC

15. 12 x 2 Step 4 Step 5 8x 2 + 14x– 10 + 16x – 3 Look for the same variable and exponent combinations Combine like terms in columns Step 6Add terms Note:When you add or subtract polynomials exponents do not change Step 2 Step 1 – ( ) means a red flag – mistake zone Add the – to each sign in the ( ) – + 4x 2 – – 2x– – 7 Step 3 Rewrite with one sign for each term – 4x 2 + 2x + 7 – (4x 2 – 2x – 7) 02/11/12lntaylor © TOC

16. Multiplying Polynomials 02/11/12lntaylor © TOC

17. – 20x 2 Step 4 Step 5 – 5x 2 + 10x– 18 + x + 3 Look for the same variable and exponent combinations Combine like terms in columns Step 6Add terms Note:When you add or subtract polynomials exponents do not change Step 2 Step 1 – ( ) means a red flag – mistake zone Multiply and then add the – to each sign in the ( ) – – 15x 2 – + 9x– – 21 Step 3 Rewrite with one sign for each term + 15x 2 – 9x + 21 – 3(– 5x 2 + 3x – 7) 02/11/12lntaylor © TOC

18. Now you try 12x 2 + 14x – 10 – 6(4x 2 – 2x – 6) 02/11/12lntaylor © TOC

19. 12x 2 Step 4 Step 5 – 12x 2 + 14x– 10 + 26 Look for the same variable and exponent combinations Combine like terms in columns Step 6Add terms Note:When you add or subtract polynomials exponents do not change Step 2 Step 1 – ( ) means a red flag – mistake zone Multiply and then add the – to each sign in the ( ) – + 24x 2 – – 12x– – 36 Step 3 Rewrite with one sign for each term – 24x 2 + 12x + 36 – 6(4x 2 – 2x – 6) 02/11/12lntaylor © TOC + 28 x

20. Multiplying binomials and trinomials (3x 2 + 4x) (4x 3 – 6) 02/11/12lntaylor © TOC

21. (3x 2 + 4x)(4x 3 – 6) Step 1 Multiply 1 st term times everything in 2 nd parenthesis 3x 2 (Multiply coefficients and add exponents) 3x 2 (4x 3 ) =+ 12x 5 3x 2 (-6) = -18x 2 Step 2Multiply 2nd term times everything in 2 nd parenthesis 4x 4x (4x 3 ) = + 16x 4 4x (-6) =-24x Step 3Add terms with the same bases and exponents Step 4Rearrange terms in descending order of exponents 12x 5 + 16x 4 – 18x 2 – 24x 02/11/12lntaylor © TOC

22. Now you try (3x 2 + 4x) (5x 4 – 8) 02/11/12lntaylor © TOC

23. (3x 2 + 4x)(5x 4 – 8) Step 1 Multiply 1 st term times everything in 2 nd parenthesis 3x 2 (Multiply coefficients and add exponents) 3x 2 (5x 4 ) =+ 15x 6 3x 2 (-8) =-24x 2 Step 2Multiply 2nd term times everything in 2 nd parenthesis 4x 4x (5x 4 ) =+ 20x 5 4x (-8) = -32x Step 3Add terms with the same bases and exponents Step 4Rearrange terms in descending order of exponents 15x 6 + 20x 5 – 24x 2 – 32x 02/11/12lntaylor © TOC

24. Special Case Binomials 02/11/12lntaylor © TOC

25. Special Case Binomials Perfect square binomials (x + 4) 2 02/11/12lntaylor © TOC

26. (x + 4) 2 Step 1 Square the first term x 2 Step 2 Multiply last term by 2x (+ 4) 2x Step 3 Square the last term + 4 2 Step 4 Rewrite the terms x 2 + 8x + 16 02/11/12lntaylor © TOC

27. Now you try (x – 8) 2 02/11/12lntaylor © TOC

28. (x – 8) 2 Step 1 Square the first term x 2 Step 2 Multiply last term by 2x (– 8) 2x Step 3 Square the last term – 8 2 Step 4 Rewrite the terms x 2 – 16x + 64 02/11/12lntaylor © TOC

29. Special Case Binomials Difference of squares binomials (x + 4) (x – 4) 02/11/12lntaylor © TOC

30. Step 1 Square the first term (x + 4) (x – 4)(x)x Step 2Multiply the last terms in each parenthesis ( + 4)(– 4) Step 3 Rewrite x 2 - 16 Step 4 Note there are only two terms 02/11/12lntaylor © TOC

31. Now you try (x + 6) (x – 6) 02/11/12lntaylor © TOC

32. Step 1 Square the first term (x + 6) (x – 6)(x)x Step 2Multiply the last terms in each parenthesis ( + 6)(– 6) Step 3 Rewrite x 2 - 36 Step 4 Note there are only two terms 02/11/12lntaylor © TOC

33. Shortcuts to Multiplying Binomials 02/11/12lntaylor © TOC

34. Shortcuts (x + 8) (x + 6) 02/11/12lntaylor © TOC

35. (x + 8) (x + 6) Step 1 Multiply the first terms in each parenthesis (x)x Step 2 Add the last terms in each parenthesis (+ 8+ 6) Step 3 Multiply by the variable!!!! x Step 4 Multiply the last terms in each parenthesis + 8(6) Step 5 Rewrite x 2 + 14x + 48 02/11/12lntaylor © TOC

36. Now you try (x + 10) (x – 4) 02/11/12lntaylor © TOC

37. (x + 10) (x – 4) Step 1 Multiply the first terms in each parenthesis (x)x Step 2 Add the last terms in each parenthesis (+ 10– 4) Step 3 Multiply by the variable!!!! x Step 4 Multiply the last terms in each parenthesis + 10(– 4) Step 5 Rewrite x 2 + 6x – 40 02/11/12lntaylor © TOC

38. Now you try (x – 8) (x – 3) 02/11/12lntaylor © TOC

39. (x – 8) (x – 3) Step 1 Multiply the first terms in each parenthesis (x)x Step 2 Add the last terms in each parenthesis (– 8– 3) Step 3 Multiply by the variable!!!! x Step 4 Multiply the last terms in each parenthesis – 8(– 3) Step 5 Rewrite x 2 – 11x + 24 02/11/12lntaylor © TOC

40. Dividing Polynomials 02/11/12lntaylor © TOC

41. Dividing Polynomials 4x 2 + 6x 2x 02/11/12lntaylor © TOC

42. 4x 2 + 6x 2x Step 1 Split the terms 4x 2 2x + 6x 2x Step 2 Simplify: reduce the coefficients and subtract the exponents 2x + 3 02/11/12lntaylor © TOC

43. Now you try 14x 4 + 6x 2 – 3x 2x 02/11/12lntaylor © TOC

44. 14x 4 + 6x 2 – 3x 2x Step 1 Split the terms 14x 4 2x + 6x 2 2x Step 2 Simplify: reduce the coefficients and subtract the exponents 7x 3 + 3x – 3x 2x – 3 2 02/11/12lntaylor © TOC

45. Factoring Polynomials 02/11/12lntaylor © TOC

46. Factoring Polynomials x 2 + 6x + 8 02/11/12lntaylor © TOC

47. x 2 + 6x + 8 Step 1 Check to see if last term is positive + Step 2 Divide middle term coefficient by 2 + 6 2 = 3 Step 3 Square the answer and check last term (3)(3) = 9 8 Step 4 If they are both the same you have your factors No match Step 5 If they are not the same subtract one and add one (2)(4) = 8 Step 6 Continue until numbers match Yes Step 7 Add an x to each parenthesis (x + 2)(x + 4) 02/11/12lntaylor © TOC

48. Now you try x 2 – 20x + 96 02/11/12lntaylor © TOC

49. x 2 – 20x + 96 Step 1 Check to see if last term is positive + Step 2 Divide middle term coefficient by 2 – 20 2 = – 10 Step 3 Square the answer and check last term (-10)(-10) = 100 96 Step 4 If they are both the same you have your factors Step 5 If they are not the same subtract one and add one (-11)(-9) = 99 Step 6 Continue until numbers match Yes (-12)(-8) = 96 Step 7 Add an x to each parenthesis (x – 12)(x – 8) 02/11/12lntaylor © TOC

50. Now you try x 2 + 15x + 56 02/11/12lntaylor © TOC

51. x 2 + 15x + 56 Step 1 Check to see if last term is positive + Step 2 Divide middle term coefficient by 2 + 15 2 = 7.5 Step 3 Since the answer includes 0.5 round up and down (8)(7) = 56 56 Step 4 If they are both the same you have your factors Step 5 If they are not the same subtract one and add one Step 6 Continue until numbers match Yes Step 7 Add an x to each parenthesis (x + 8)(x + 7) 02/11/12lntaylor © TOC

52. Now you try x 2 – 19x + 84 02/11/12lntaylor © TOC

53. x 2 – 19x + 84 Step 1 Check to see if last term is positive + Step 2 Divide middle term coefficient by 2 – 19 2 = – 9.5 Step 3 Since the answer includes 0.5 round up and down (-10)(-9) = 90 84 Step 4 If they are both the same you have your factors Step 5 If they are not the same subtract one and add one (-11)(-8) = 88 Step 6 Continue until numbers match Yes (-12)(-7) = 84 Step 7 Add an x to each parenthesis (x – 12)(x – 7) 02/11/12lntaylor © TOC

54. Factoring Negative Constants x 2 – 2x – 48 02/11/12lntaylor © TOC

55. x 2 – 2x – 48 Step 1 Check to see if last term is negative – Step 2 Construct a table for the “difference” of factors for the constant 22422 31613 4128 682 Step 3 Match the middle term coefficient to the last column 2 Step 4 Use the numbers in the 1 st and 2 nd columns; add a variable to each (x 6) (x 8) Step 5 Put the middle term sign next to the largest number – Step 6 Put the opposite sign next to the smallest number!!!! + 02/11/12lntaylor © TOC

56. Now you try x 2 – 11x – 12 02/11/12lntaylor © TOC

57. x 2 – 11x – 12 Step 1 Check to see if last term is negative – Step 2 Construct a table for the “difference” of factors for the constant 11211 264 341 Step 3 Match the middle term coefficient to the last column 11 Step 4 Use the numbers in the 1 st and 2 nd columns; add a variable to each (x 1) (x 12) Step 5 Put the middle term sign next to the largest number – Step 6 Put the opposite sign next to the smallest number!!!! + 02/11/12lntaylor © TOC

58. Special cases x 2 – 49 02/11/12lntaylor © TOC

59. x 2 – 49 Step 1 There are only two terms and the last term is negative Step 2 Square root the last term 49 ½ =± 7 Step 3 Write one factor with + and one factor with – (x + 7) (x – 7) 02/11/12lntaylor © TOC

60. Now you try x 2 – 225 02/11/12lntaylor © TOC

61. x 2 – 225 Step 1 There are only two terms and the last term is negative Step 2 Square root the last term 225 ½ =± 15 Step 3 Write one factor with + and one factor with – (x + 15) (x – 15) 02/11/12lntaylor © TOC

62. Super duper complicated problems x 2 – 7x + 12 x – 4 02/11/12lntaylor © TOC

63. Super duper complicated problems – Not!!!!! x 2 – 7x + 12 x – 4 Step 1 These are not really complicated!!! Step 2 Check if constant in the denominator divides the constant in the numerator + 12 – 4 – 3= Step 3 Add an x to get the answer x – 3 02/11/12lntaylor © TOC

64. Super duper complicated problem you can try x 2 – 2x – 35 x + 5 02/11/12lntaylor © TOC

65. Super duper complicated problems – Not!!!!! x 2 – 2x – 35 x + 5 Step 1 These are not really complicated!!! Step 2 Check if constant in the denominator divides the constant in the numerator – 35 + 5 – 7= Step 3 Add an x to get the answer x – 7 02/11/12lntaylor © TOC