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I have a huge polynomial, and I am having the following issues 1. I wanted to find the largest and smallest degree of that polynomial. 2. How to truncate the lower order terms, sometimes higher order terms and some times interested in only extracting the middle order terms. how to achieve this? 3. I wanted to analyze how the accuracy of the roots of that polynomial are going to change only when only lower, middle and higher order terms are considered? 4. Bit difficulty in finding the roots of this equation using NSolve.

P=-(9.47789*10^249/((1.31057*10^6 + 0.719881 x^2)^4 (-4.9348*10^7 + 
          78.5 x^2)^3 (7.00903*10^10 - 1.5772*10^6 x^2 + 
          7.9463 x^4)^3 (1.57285*10^24 - 3.69325*10^19 x^2 + 
          2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
          149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
         6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
         4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 
         7.20343*10^6 x^14 + 
         21.6477 x^16))) + (1.77008*10^246 x^2)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) - (1.54404*10^242 x^4)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) + (8.35686*10^237 x^6)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) - (3.14421*10^233 x^8)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) + (8.73169*10^228 x^10)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) - (1.85476*10^224 x^12)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) + (3.08146*10^219 x^14)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) - (4.05986*10^214 x^16)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) + (4.27501*10^209 x^18)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) - (3.60829*10^204 x^20)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) + (2.43721*10^199 x^22)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) - (1.30869*10^194 x^24)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) + (5.51338*10^188 x^26)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) - (1.77981*10^183 x^28)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) + (4.20981*10^177 x^30)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) - (6.57836*10^171 x^32)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) + (4.46704*10^165 x^34)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) + (5.82581*10^159 x^36)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) - (1.70088*10^154 x^38)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) + (1.18042*10^148 x^40)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) + (1.0515*10^142 x^42)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) - (2.09243*10^136 x^44)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) + (3.05662*10^129 x^46)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) + (1.44122*10^124 x^48)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) - (6.9849*10^117 x^50)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) - (5.58488*10^111 x^52)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) + (3.95955*10^105 x^54)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) + (1.34018*10^99 x^56)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) - (1.16083*10^93 x^58)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) - (1.93372*10^86 x^60)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) + (1.87434*10^80 x^62)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) + (1.17814*10^73 x^64)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) - (1.56905*10^67 x^66)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) + (5.37283*10^59 x^68)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) + (5.15634*10^53 x^70)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) - (7.60002*10^46 x^72)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) + (4.21574*10^39 x^74)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) - (1.0729*10^32 x^76)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) + (1.22052*10^24 x^78)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16)) - (4.73979*10^15 x^80)/((1.31057*10^6 + 
       0.719881 x^2)^4 (-4.9348*10^7 + 78.5 x^2)^3 (7.00903*10^10 - 
       1.5772*10^6 x^2 + 7.9463 x^4)^3 (1.57285*10^24 - 
       3.69325*10^19 x^2 + 2.11644*10^14 x^4 - 1.44848*10^8 x^6 - 
       149.684 x^8)^2 (8.12192*10^45 - 3.73597*10^41 x^2 + 
      6.30119*10^36 x^4 - 4.66731*10^31 x^6 + 1.35431*10^26 x^8 - 
      4.19872*10^19 x^10 - 1.30887*10^14 x^12 + 7.20343*10^6 x^14 + 
      21.6477 x^16))
$\endgroup$
  • 2
    $\begingroup$ Your P is not a polynomial. As a first step, just FullSimplify[P] already gives a much simpler expression: it's a ratio of two polynomials, where the numerator is a degree-80 polynomial and the denominator is a degree-58 polynomial. Have a look at A = FullSimplify[P] and then Numerator[A] and Denominator[A]. For finding roots it's enough to look for roots of the numerator: Solve[Numerator[A] == 0, x]. $\endgroup$ – Roman May 27 at 14:03
  • $\begingroup$ But I have advised in this multiple times that do not use FullSimiplify, as you know it takes to much time. $\endgroup$ – acoustics May 27 at 14:08
  • $\begingroup$ You have been given poor advice then. $\endgroup$ – Roman May 27 at 14:09
  • $\begingroup$ And about my 2, 3 and 4 questions, you have any suggestions? $\endgroup$ – acoustics May 27 at 14:10
  • $\begingroup$ See below: use Solve on the numerator. $\endgroup$ – Roman May 27 at 14:11
2
$\begingroup$
A = FullSimplify[P]

(-2.9978*10^236 + 5.59865*10^232 x^2 - 4.8837*10^228 x^4 + 2.64322*10^224 x^6 - 9.94494*10^219 x^8 + 2.76178*10^215 x^10 - 5.86649*10^210 x^12 + 9.74647*10^205 x^14 - 1.28411*10^201 x^16 + 1.35216*10^196 x^18 - 1.14128*10^191 x^20 + 7.70875*10^185 x^22 - 4.13931*10^180 x^24 + 1.74385*10^175 x^26 - 5.62943*10^169 x^28 + 1.33154*10^164 x^30 - 2.0807*10^158 x^32 + 1.4129*10^152 x^34 + 1.84267*10^146 x^36 - 5.37978*10^140 x^38 + 3.7336*10^134 x^40 + 3.32583*10^128 x^42 - 6.61823*10^122 x^44 + 9.6679*10^115 x^46 + 4.55849*10^110 x^48 - 2.20928*10^104 x^50 - 1.76646*10^98 x^52 + 1.25238*10^92 x^54 + 4.23891*10^85 x^56 - 3.67163*10^79 x^58 - 6.11624*10^72 x^60 + 5.92842*10^66 x^62 + 3.72638*10^59 x^64 - 4.96281*10^53 x^66 + 1.69939*10^46 x^68 + 1.63092*10^40 x^70 - 2.40384*10^33 x^72 + 1.33341*10^26 x^74 - 3.39352*10^18 x^76 + 3.86043*10^10 x^78 - 149.917 x^80)/((1.82054*10^6 + 1. x^2)^4 (-5.54489*10^15 + 1.33594*10^11 x^2 - 827119. x^4 + 1. x^6)^3 (1.05078*10^22 - 2.46736*10^17 x^2 + 1.41394*10^12 x^4 - 967692. x^6 - 1. x^8)^2 (3.75186*10^44 - 1.7258*10^40 x^2 + 2.91079*10^35 x^4 - 2.15603*10^30 x^6 + 6.25614*10^24 x^8 - 1.93957*10^18 x^10 - 6.04623*10^12 x^12 + 332757. x^14 + 1. x^16))

Solve[Numerator[A] == 0, x, Reals]

{{x -> -11270.4}, {x -> -7158.29}, {x -> -6205.96}, {x -> -4872.44}, {x -> -3463.08}, {x -> -2463.91}, {x -> -1831.79}, {x -> -1502.94}, {x -> -1278.71}, {x -> -533.326}, {x -> 533.326}, {x -> 1278.71}, {x -> 1502.94}, {x -> 1831.79}, {x -> 2463.91}, {x -> 3463.08}, {x -> 4872.44}, {x -> 6205.96}, {x -> 7158.29}, {x -> 11270.4}}

Truncation: if we only take the polynomial terms up to $x^{10}$, for example,

Solve[Normal[Series[Numerator[A], {x, 0, 10}]] == 0, x, Reals]

{{x -> -113.45}, {x -> 113.45}}

$\endgroup$
  • $\begingroup$ If I change the {x,0,10} to {x,0,20} I may get an entirely different set of roots right?, Actually I am trying to find the simple expression so the upon solving I should approximately get the same roots using full expression $\endgroup$ – acoustics May 27 at 14:16
  • 1
    $\begingroup$ Correct. And as @DanielLichtblau commented above, you can try series-expanding around infinity instead of zero. $\endgroup$ – Roman May 27 at 14:23
  • $\begingroup$ What benefit we achieve if series-expanding around the infinity rather than zero. $\endgroup$ – acoustics May 27 at 14:25
  • 1
    $\begingroup$ @acoustics just try it, play with it, and see what happens to the roots. You can also hire a professional consultant to guide you. $\endgroup$ – Roman May 27 at 14:39

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