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Bot GPT 3.5 To find the roots of the polynomial equation 2x^3 + 2x^2 - 19x + 20 = 0, we can use various methods such as factoring, synthetic division, or numerical methods.
One method to find the roots is factoring by grouping.
2x^3 + 2x^2 - 19x + 20 = 0
Rearranging the terms:
2x^3 - 19x + 2x^2 + 20 = 0
Grouping the terms:
(x^2 + 10)(2x - 1) = 0
Setting each factor equal to zero:
x^2 + 10 = 0 OR 2x - 1 = 0
Solving the first equation:
x^2 + 10 = 0
x^2 = -10
x = ± √(-10)
Since there is no real number that satisfies the equation x^2 = -10, this quadratic factor does not have real solutions.
Solving the second equation:
2x - 1 = 0
2x = 1
x = 1/2
Therefore, the only real root of the polynomial equation 2x^3 + 2x^2 - 19x + 20 = 0 is x = 1/2.