Question
Determine the nature of the roots of the polynomial equation 2x^3-5x^2+5x-2=0
Answers
GPT 3.5
To determine the nature of the roots of the polynomial equation 2x^3 - 5x^2 + 5x - 2 = 0, we can use the discriminant.
The discriminant (D) of a polynomial equation of the form ax^3 + bx^2 + cx + d = 0 is given by:
D = (18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2)
For the equation 2x^3 - 5x^2 + 5x - 2 = 0, we have:
a = 2
b = -5
c = 5
d = -2
D = (18 * 2 * (-5) * 5 * (-2) - 4 * (-5)^3 * (-2) + (-5)^2 * 5^2 - 4 * 2 * 5^3 - 27 * 2^2 * (-2)^2)
= (-1800 + 2000 + 625 - 2000 - 216)
= 409
Since the discriminant is positive (D > 0), the equation has three distinct real roots.
The discriminant (D) of a polynomial equation of the form ax^3 + bx^2 + cx + d = 0 is given by:
D = (18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2)
For the equation 2x^3 - 5x^2 + 5x - 2 = 0, we have:
a = 2
b = -5
c = 5
d = -2
D = (18 * 2 * (-5) * 5 * (-2) - 4 * (-5)^3 * (-2) + (-5)^2 * 5^2 - 4 * 2 * 5^3 - 27 * 2^2 * (-2)^2)
= (-1800 + 2000 + 625 - 2000 - 216)
= 409
Since the discriminant is positive (D > 0), the equation has three distinct real roots.