Algorithms in Solving Polynomial Inequalities

A new method to solve the solution set of polynomial inequalities was conducted. When 𝑥 − 𝑟1 𝑥 − 𝑟2 > 0 𝑤ℎ𝑒𝑟𝑒 𝑟1, 𝑟2 ∈ ℝ 𝑎𝑛𝑑 𝑟1 < 𝑟2, the solution set is 𝑥 ∈ ℝ 𝑥 ∈ −∞, 𝑟2 ∪ 𝑟1, ∞ }. Thus, when the inequality is (𝑥 − 𝑟1) 𝑥 − 𝑟2 ≥ 0, then the solution set is 𝑥 ∈ ℝ 𝑥 ∈ −∞, 𝑟1 ∪ [𝑟2, ∞). If 𝑥 − 𝑟1 𝑥 − 𝑟2 < 0, then the solution set is 𝑥 ∈ ℝ 𝑥 ∈ 𝑟1, 𝑟2 }. Thus when 𝑥 − 𝑟1 𝑥 − 𝑟2 ≤ 0, the solution set is 𝑥 ∈ ℝ 𝑥 ∈ [𝑟1, 𝑟2] }. Let 𝑓 𝑥 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 where 𝑎 ≠ 0, 𝑏 𝑎𝑛𝑑 𝑐 ∈ ℝ. If 𝑏2 − 4𝑎𝑐 < 0, then the solution of quadratic inequalities is {ℝ} when, by substitution of a particular real number, the inequality is true. Otherwise, the solution of the inequality is ∅. Let 𝑟1 < 𝑟2 < . . . < 𝑟𝑛 ∈ ℝ and 𝑛 ≥ 3. Let 𝑥 − 𝑟1 𝑥 − 𝑟2 … 𝑥 − 𝑟𝑛 > 0 if n is even. Then, the solution set is 𝑥 ∈ ℝ 𝑥 ∈ −∞, 𝑟1 ∪ 𝑟𝑛 , +∞ ∪ 𝑟𝑖 , 𝑟𝑖+1 : 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 }. Thus, when 𝑥 − 𝑟1𝑥−𝑟2…𝑥−𝑟𝑛≥0, the solution is 𝑥 ∈ ℝ 𝑥 ∈−∞, 𝑟1∪𝑟𝑛 ,+∞∪𝑟𝑖, 𝑟𝑖+1: 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 }. If 𝑛 is odd, then the solution set is 𝑥 ∈ ℝ 𝑥 ∈ 𝑟𝑛 , +∞ ∪ 𝑟𝑖 , 𝑟𝑖+1 : 𝑖 𝑖𝑠 𝑜𝑑𝑑 }. Thus, when 𝑥 − 𝑟1 𝑥 − 𝑟2…𝑥−𝑟𝑛≥0, the solution set is 𝑥 ∈ ℝ 𝑥 ∈𝑟𝑛 ,+∞∪𝑟𝑖, 𝑟𝑖+1:𝑖 𝑖𝑠 𝑜𝑑𝑑 }. Let 𝑥−𝑟1𝑥−𝑟2…𝑥−𝑟𝑛<0 if n is even. Then, the solution set is 𝑥 ∈ ℝ 𝑥 ∈ 𝑟𝑖 , 𝑟𝑖+1 ∶ 𝑖 𝑖𝑠 𝑜𝑑𝑑 }. Thus, when 𝑥 − 𝑟1 𝑥 − 𝑟2…𝑥−𝑟𝑛≤0, then the solution set is 𝑥 ∈ ℝ 𝑥 ∈𝑟𝑖, 𝑟𝑖+1:𝑖 𝑖𝑠 𝑜𝑑𝑑 }. If 𝑛 is an odd, then the solution set is 𝑥 ∈ ℝ 𝑥 ∈ −∞, 𝑟1 ∪ 𝑟𝑖 , 𝑟𝑖+1 : 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 }. Thus, when 𝑥 − 𝑟1 𝑥 − 𝑟2 … 𝑥 − 𝑟𝑛 ≤ 0, the solution set is 𝑥 ∈ ℝ 𝑥 ∈ −∞, 𝑟1 ∪ 𝑟𝑖 , 𝑟𝑖+1 : 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 }. This research provides a novel method in solving the solution set of polynomial inequalities, in addition to other existing methods.

Keywords: polynomial, polynomial inequality, solution, solution set, quadratic inequality, real roots and imaginary roots