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Simplifying Algebraic Fractions (Two Terms to Single Fraction)

Simplifying Algebraic Fractions (Two Terms to Single Fraction)
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2019-10-06 17:43
Geometric Construction
This example demonstrates how to combine two or more algebraic fractions into a single fraction. Core math concepts include fraction simplification, Least Common Multiple (LCM), Greatest Common Divisor (GCD), etc. GeoGebra commands like If, LCM, Simplify, GCD, Remove, TextToUnicode are used, along with objects such as simpler, counter, G5 to M4. Suitable for high school mathematics teaching to reinforce students' fraction operation skills.

GeoGebra Commands & Parameters

  1. If(counter < simpler, RandomElement[{-1, 1, -2, 2}], (RandomElement[{-1, 1}] * Random[2, 5])) → G5
  2. If(counter < simpler + 2, Random[2, 7], Random[2, 12]) → D5
  3. If(counter < simpler + 1, D5, RandomElement[Remove[{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, {D5}]]) → I5
  4. If(abs(A4) ≟ 1, If[A4 < 0, (-Simplify[(B4 * x) + C4]), Simplify[(B4 * x) + C4]], (A4 * Simplify[(B4 * x) + C4])) → J4
  5. If(abs(F4) ≟ 1, If[F4 < 0, (-Simplify[(G4 * x) + H4]), Simplify[(G4 * x) + H4]], (F4 * Simplify[(G4 * x) + H4])) → K4
  6. If(abs(A3) ≟ 1, If[A3 < 0, (-Simplify[(B3 * x) + C3]), Simplify[(B3 * x) + C3]], (A3 * Simplify[(B3 * x) + C3])) → J3
  7. If(abs(F3) ≟ 1, If[F3 < 0, (-Simplify[(G3 * x) + H3]), Simplify[(G3 * x) + H3]], (F3 * Simplify[(G3 * x) + H3])) → K3
  8. If(abs(A2) ≟ 1, If[A2 < 0, (-Simplify[(B2 * x) + C2]), Simplify[(B2 * x) + C2]], (A2 * Simplify[(B2 * x) + C2])) → J2
  9. If(abs(F2) ≟ 1, If[F2 < 0, (-Simplify[(G2 * x) + H2]), Simplify[(G2 * x) + H2]], (F2 * Simplify[(G2 * x) + H2])) → K2
  10. If(abs(A1) ≟ 1, If[A1 < 0, (-Simplify[(B1 * x) + C1]), Simplify[(B1 * x) + C1]], (A1 * Simplify[(B1 * x) + C1])) → J1
  11. If(abs(F1) ≟ 1, If[F1 < 0, (-Simplify[(G1 * x) + H1]), Simplify[(G1 * x) + H1]], (F1 * Simplify[(G1 * x) + H1])) → K1
  12. If(E1 > 0, L1(x) + M1(x), L1(x) - M1(x)) → N1
  13. If(E2 > 0, L2(x) + M2(x), L2(x) - M2(x)) → N2
  14. If(E3 > 0, L3(x) + M3(x), L3(x) - M3(x)) → N3
  15. If(E4 > 0, L4(x) + M4(x), L4(x) - M4(x)) → N4
  16. LCM(D1, I1) → O1
  17. LCM(D2, I2) → O2
  18. LCM(D3, I3) → O3
  19. LCM(D4, I4) → O4
  20. If(E4 > 0, If[abs((A4 * O4 / D4)) ≟ 1, If[(A4 * O4 / D4) > 0, Simplify[(B4 * x) + C4], (-Simplify[(B4 * x) + C4])], (Simplify[(A4 * O4 / D4)] * Simplify[(B4 * x) + C4])] + If[abs((F4 * O4 / I4)) ≟ 1, If[(F4 * O4 / I4) > 0, Simplify[(G4 * x) + H4], (-Simplify[(G4 * x) + H4])], (Simplify[(F4 * O4 / I4)] * Simplify[(G4 * x) + H4])], If[abs((A4 * O4 / D4)) ≟ 1, If[(A4 * O4 / D4) > 0, Simplify[(B4 * x) + C4], (-Simplify[(B4 * x) + C4])], (Simplify[(A4 * O4 / D4)] * Simplify[(B4 * x) + C4])] - If[abs((F4 * O4 / I4)) ≟ 1, If[(F4 * O4 / I4) > 0, Simplify[(G4 * x) + H4], (-Simplify[(G4 * x) + H4])], (Simplify[(F4 * O4 / I4)] * Simplify[(G4 * x) + H4])]) → P4
  21. If(E2 > 0, If[abs((A2 * O2 / D2)) ≟ 1, If[(A2 * O2 / D2) > 0, Simplify[(B2 * x) + C2], (-Simplify[(B2 * x) + C2])], (Simplify[(A2 * O2 / D2)] * Simplify[(B2 * x) + C2])] + If[abs((F2 * O2 / I2)) ≟ 1, If[(F2 * O2 / I2) > 0, Simplify[(G2 * x) + H2], (-Simplify[(G2 * x) + H2])], (Simplify[(F2 * O2 / I2)] * Simplify[(G2 * x) + H2])], If[abs((A2 * O2 / D2)) ≟ 1, If[(A2 * O2 / D2) > 0, Simplify[(B2 * x) + C2], (-Simplify[(B2 * x) + C2])], (Simplify[(A2 * O2 / D2)] * Simplify[(B2 * x) + C2])] - If[abs((F2 * O2 / I2)) ≟ 1, If[(F2 * O2 / I2) > 0, Simplify[(G2 * x) + H2], (-Simplify[(G2 * x) + H2])], (Simplify[(F2 * O2 / I2)] * Simplify[(G2 * x) + H2])]) → P2
  22. If(E3 > 0, If[abs((A3 * O3 / D3)) ≟ 1, If[(A3 * O3 / D3) > 0, Simplify[(B3 * x) + C3], (-Simplify[(B3 * x) + C3])], (Simplify[(A3 * O3 / D3)] * Simplify[(B3 * x) + C3])] + If[abs((F3 * O3 / I3)) ≟ 1, If[(F3 * O3 / I3) > 0, Simplify[(G3 * x) + H3], (-Simplify[(G3 * x) + H3])], (Simplify[(F3 * O3 / I3)] * Simplify[(G3 * x) + H3])], If[abs((A3 * O3 / D3)) ≟ 1, If[(A3 * O3 / D3) > 0, Simplify[(B3 * x) + C3], (-Simplify[(B3 * x) + C3])], (Simplify[(A3 * O3 / D3)] * Simplify[(B3 * x) + C3])] - If[abs((F3 * O3 / I3)) ≟ 1, If[(F3 * O3 / I3) > 0, Simplify[(G3 * x) + H3], (-Simplify[(G3 * x) + H3])], (Simplify[(F3 * O3 / I3)] * Simplify[(G3 * x) + H3])]) → P3
  23. If(E1 > 0, If[abs((A1 * O1 / D1)) ≟ 1, If[(A1 * O1 / D1) > 0, Simplify[(B1 * x) + C1], (-Simplify[(B1 * x) + C1])], (Simplify[(A1 * O1 / D1)] * Simplify[(B1 * x) + C1])] + If[abs((F1 * O1 / I1)) ≟ 1, If[(F1 * O1 / I1) > 0, Simplify[(G1 * x) + H1], (-Simplify[(G1 * x) + H1])], (Simplify[(F1 * O1 / I1)] * Simplify[(G1 * x) + H1])], If[abs((A1 * O1 / D1)) ≟ 1, If[(A1 * O1 / D1) > 0, Simplify[(B1 * x) + C1], (-Simplify[(B1 * x) + C1])], (Simplify[(A1 * O1 / D1)] * Simplify[(B1 * x) + C1])] - If[abs((F1 * O1 / I1)) ≟ 1, If[(F1 * O1 / I1) > 0, Simplify[(G1 * x) + H1], (-Simplify[(G1 * x) + H1])], (Simplify[(F1 * O1 / I1)] * Simplify[(G1 * x) + H1])]) → P1
  24. If(E3 > 0, Expand[Simplify[(Simplify[(A3 * O3 / D3)] * Simplify[(B3 * x) + C3])]] + Expand[Simplify[(Simplify[(F3 * O3 / I3)] * Simplify[(G3 * x) + H3])]], Expand[Simplify[(Simplify[(A3 * O3 / D3)] * Simplify[(B3 * x) + C3])]] - Expand[Simplify[(Simplify[(F3 * O3 / I3)] * Simplify[(G3 * x) + H3])]]) → R3
  25. If(E4 > 0, Expand[Simplify[(Simplify[(A4 * O4 / D4)] * Simplify[(B4 * x) + C4])]] + Expand[Simplify[(Simplify[(F4 * O4 / I4)] * Simplify[(G4 * x) + H4])]], Expand[Simplify[(Simplify[(A4 * O4 / D4)] * Simplify[(B4 * x) + C4])]] - Expand[Simplify[(Simplify[(F4 * O4 / I4)] * Simplify[(G4 * x) + H4])]]) → R4
  26. If(counter < 3 ∧ G5 < 0, 1, (Mod[counter, 2] * Mod[I5, 2])) → E5
  27. If(E1 > 0, Expand[Simplify[(Simplify[(A1 * O1 / D1)] * Simplify[(B1 * x) + C1])]] + Expand[Simplify[(Simplify[(F1 * O1 / I1)] * Simplify[(G1 * x) + H1])]], Expand[Simplify[(Simplify[(A1 * O1 / D1)] * Simplify[(B1 * x) + C1])]] - Expand[Simplify[(Simplify[(F1 * O1 / I1)] * Simplify[(G1 * x) + H1])]]) → R1
  28. If(E2 > 0, Expand[Simplify[(Simplify[(A2 * O2 / D2)] * Simplify[(B2 * x) + C2])]] + Expand[Simplify[(Simplify[(F2 * O2 / I2)] * Simplify[(G2 * x) + H2])]], Expand[Simplify[(Simplify[(A2 * O2 / D2)] * Simplify[(B2 * x) + C2])]] - Expand[Simplify[(Simplify[(F2 * O2 / I2)] * Simplify[(G2 * x) + H2])]]) → R2
  29. Simplify(R1) → S1
  30. Simplify(R2) → S2
  31. Simplify(R3) → S3
  32. Simplify(R4) → S4
  33. GCD(Join[{Coefficients[S3], {O3}}]) → U3
  34. GCD(Join[{Coefficients[S4], {O4}}]) → U4
  35. GCD(Join[{Coefficients[S2], {O2}}]) → U2
  36. GCD(Join[{Coefficients[S1], {O1}}]) → U1
  37. If(counter < simpler, 1, Random[2, 3]) → A5
  38. If(counter < simpler, Random[1, 3], (RandomElement[{-1, 1, 1, 1}] * Random[3, 7])) → C5
  39. If(counter < simpler, 1, (RandomElement[{-1, 1, 1, 1}] * Random[2, 3])) → B5
  40. If(abs(A5) ≟ 1, If[A5 < 0, (-Simplify[(B5 * x) + C5]), Simplify[(B5 * x) + C5]], (A5 * Simplify[(B5 * x) + C5])) → J5
  41. If(counter < simpler, 1, Random[2, 3]) → F5
  42. If(abs(F5) ≟ 1, If[F5 < 0, (-Simplify[(G5 * x) + H5]), Simplify[(G5 * x) + H5]], (F5 * Simplify[(G5 * x) + H5])) → K5
  43. If(E5 > 0, L5(x) + M5(x), L5(x) - M5(x)) → N5
  44. LCM(D5, I5) → O5
  45. If(E5 > 0, If[abs((A5 * O5 / D5)) ≟ 1, If[(A5 * O5 / D5) > 0, Simplify[(B5 * x) + C5], (-Simplify[(B5 * x) + C5])], (Simplify[(A5 * O5 / D5)] * Simplify[(B5 * x) + C5])] + If[abs((F5 * O5 / I5)) ≟ 1, If[(F5 * O5 / I5) > 0, Simplify[(G5 * x) + H5], (-Simplify[(G5 * x) + H5])], (Simplify[(F5 * O5 / I5)] * Simplify[(G5 * x) + H5])], If[abs((A5 * O5 / D5)) ≟ 1, If[(A5 * O5 / D5) > 0, Simplify[(B5 * x) + C5], (-Simplify[(B5 * x) + C5])], (Simplify[(A5 * O5 / D5)] * Simplify[(B5 * x) + C5])] - If[abs((F5 * O5 / I5)) ≟ 1, If[(F5 * O5 / I5) > 0, Simplify[(G5 * x) + H5], (-Simplify[(G5 * x) + H5])], (Simplify[(F5 * O5 / I5)] * Simplify[(G5 * x) + H5])]) → P5
  46. If(E5 > 0, Expand[Simplify[(Simplify[(A5 * O5 / D5)] * Simplify[(B5 * x) + C5])]] + Expand[Simplify[(Simplify[(F5 * O5 / I5)] * Simplify[(G5 * x) + H5])]], Expand[Simplify[(Simplify[(A5 * O5 / D5)] * Simplify[(B5 * x) + C5])]] + Expand[(-1 * Simplify[(Simplify[(F5 * O5 / I5)] * Simplify[(G5 * x) + H5])])]) → R5
  47. Simplify(R5) → S5
  48. GCD(Join[{Coefficients[S5], {O5}}]) → U5
  49. Remove({2, 3, 4}, {2, 4}) → l1
  50. If(O5 / U5 ≟ 1, Simplify[S5(x) / U5], Simplify[S5(x) / U5] / Simplify[O5 / U5]) → V5
  51. If(O1 / U1 ≟ 1, Simplify[S1(x) / U1], Simplify[S1(x) / U1] / Simplify[O1 / U1]) → V1
  52. If(O2 / U2 ≟ 1, Simplify[S2(x) / U2], Simplify[S2(x) / U2] / Simplify[O2 / U2]) → V2
  53. If(O3 / U3 ≟ 1, Simplify[S3(x) / U3], Simplify[S3(x) / U3] / Simplify[O3 / U3]) → V3
  54. If(O4 / U4 ≟ 1, Simplify[S4(x) / U4], Simplify[S4(x) / U4] / Simplify[O4 / U4]) → V4
  55. If(abs(A6) ≟ 1, If[A6 < 0, (-Simplify[(B6 * x) + C6]), Simplify[(B6 * x) + C6]], (A6 * Simplify[(B6 * x) + C6])) → J6
  56. If(abs(F6) ≟ 1, If[F6 < 0, (-Simplify[(G6 * x) + H6]), Simplify[(G6 * x) + H6]], (F6 * Simplify[(G6 * x) + H6])) → K6
  57. If(E6 > 0, L6(x) + M6(x), L6(x) - M6(x)) → N6
  58. LCM(D6, I6) → O6
  59. If(E6 > 0, If[abs((A6 * O6 / D6)) ≟ 1, If[(A6 * O6 / D6) > 0, Simplify[(B6 * x) + C6], (-Simplify[(B6 * x) + C6])], (Simplify[(A6 * O6 / D6)] * Simplify[(B6 * x) + C6])] + If[abs((F6 * O6 / I6)) ≟ 1, If[(F6 * O6 / I6) > 0, Simplify[(G6 * x) + H6], (-Simplify[(G6 * x) + H6])], (Simplify[(F6 * O6 / I6)] * Simplify[(G6 * x) + H6])], If[abs((A6 * O6 / D6)) ≟ 1, If[(A6 * O6 / D6) > 0, Simplify[(B6 * x) + C6], (-Simplify[(B6 * x) + C6])], (Simplify[(A6 * O6 / D6)] * Simplify[(B6 * x) + C6])] - If[abs((F6 * O6 / I6)) ≟ 1, If[(F6 * O6 / I6) > 0, Simplify[(G6 * x) + H6], (-Simplify[(G6 * x) + H6])], (Simplify[(F6 * O6 / I6)] * Simplify[(G6 * x) + H6])]) → P6
  60. If(E6 > 0, Expand[Simplify[(Simplify[(A6 * O6 / D6)] * Simplify[(B6 * x) + C6])]] + Expand[Simplify[(Simplify[(F6 * O6 / I6)] * Simplify[(G6 * x) + H6])]], Expand[Simplify[(Simplify[(A6 * O6 / D6)] * Simplify[(B6 * x) + C6])]] - Expand[Simplify[(Simplify[(F6 * O6 / I6)] * Simplify[(G6 * x) + H6])]]) → R6
  61. Simplify(R6) → S6
  62. GCD(Join[{Coefficients[S6], {O6}}]) → U6
  63. If(O6 / U6 ≟ 1, Simplify[S6(x) / U6], Simplify[S6(x) / U6] / Simplify[O6 / U6]) → V6
  64. TextToUnicode(Text[If[type < 2, R3(x) / O3, If[type < 3, R4(x) / O4, R5(x) / O5]]]) → ListR
  65. TextToUnicode(Text[If[type < 2, Q3, If[type < 3, Q4, Q5]]]) → ListQ