Determine the nature of the roots of the polynomial equation 2x^3-5x^2+5x-2=0
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.
To determine the nature of the roots of the polynomial equation 2x^3 - 5x^2 + 5x - 2 = 0, we can use the discriminant method.
Step 1: Write the equation in the form ax^3 + bx^2 + cx + d = 0.
The given equation is already in the correct form: 2x^3 - 5x^2 + 5x - 2 = 0.
Step 2: Identify the values of a, b, c, and d.
In this case, a = 2, b = -5, c = 5, and d = -2.
Step 3: Calculate the discriminant using the formula D = b^2 - 4ac.
Plugging in the values, we get D = (-5)^2 - 4(2)(5) = 25 - 40 = -15.
Step 4: Determine the nature of the roots based on the discriminant value.
If the discriminant (D) is positive, the equation has two distinct real roots.
If the discriminant (D) is zero, the equation has a repeated real root.
If the discriminant (D) is negative, the equation has two complex conjugate roots.
In this case, the discriminant is negative (D = -15), so the polynomial equation 2x^3 - 5x^2 + 5x - 2 = 0 has two complex conjugate roots.
To determine the nature of the roots of a polynomial equation, we can use the Discriminant formula. The discriminant is the expression inside the square root in the quadratic formula and can be used to determine if the roots are real or complex. For a cubic equation, the discriminant can also provide information about the types of roots.
For a cubic equation in the form of Ax^3 + Bx^2 + Cx + D = 0, the discriminant can be calculated as follows:
Δ = 18ABCD - 4B^3D + B^2C^2 - 4AC^3 - 27A^2D^2.
Let's calculate the discriminant for the given equation: 2x^3 - 5x^2 + 5x - 2 = 0.
A = 2, B = -5, C = 5, and D = -2.
Δ = 18(2)(-5)(5)(-2) - 4(-5)^3(-2) + (-5)^2(5)^2 - 4(2)(5)^3 - 27(2)^2(-2)^2
= 1800 + 200 - 625 + 2000 - 1080
= 3295.
Now, let's analyze the value of Δ to determine the nature of the roots:
1. If Δ > 0, then the equation has three distinct real roots.
2. If Δ = 0, then the equation has at least two equal roots (either three real roots or one real and two complex conjugate roots).
3. If Δ < 0, then the equation has one real root and two complex conjugate roots.
In our case, Δ = 3295, which is greater than 0. Therefore, the given cubic equation 2x^3 - 5x^2 + 5x - 2 = 0 has three distinct real roots.