given that z1 2 and z2 3 i are roots of the cubic equation z3 bz2 cz d 0 where b c d r a write d

Step 1: Since b, c, d are real and z2 = sqrt(3) + i is nonreal, its complex conjugate is also a

Given that z1 2 and z2 3 i are roots of the cubic equation...

Given that z₁ = -2 and z₂ = √3 + i are roots of the cubic equation z³ + bz² + cz + d = 0, where b, c, d ∈ ℝ, a) write down the third root, z₃, of the equation; b) find the values of b, c, and d; 4. Given that z₁ = -2 and z₂ = √3 + i are roots of the cubic equation z + bz + cz + d = 0, where b, c, d ∈ ℝ, write down the third root, z₃, of the equation and find the values of b, c, and d.

Transcript

00:01 In this question, we have been given that alpha, beta, gamma are the roots of this equation, which is cubic, x -cube minus 2x squared minus 4x plus 5 equals to 0.

00:13 It has three roots, okay, alpha, beta and gamma.

00:18 In the first part, we have to find out the value of alpha plus beta plus gamma.

00:25 So let me write down a standard formula that we are going to use here.

00:30 Okay so with that we can easily solve this question so first of all we know that if we have a quadratic cubic equation a x cube b x square plus cx plus d is equal to zero then we know that and it is having alpha -veta gamma as root so sum is given by minus b over a then product taken two at a time is given by c over a and the product of these three three roots is given by minus d over a okay so in this case if we compare so here let me write down all those values a is one b is minus two c is minus four and d is five okay so alpha plus beta plus gamma alpha plus gamma it is equals to what minus b over a so minus of b is minus two over a which is 1 so it will be simply 2 correct then we have to find out alpha beta plus beta gamma gamma and this is equal to c divided by a so c is minus 4 a is 1 so it comes out to be minus 4 then in the next part we have to find the product of them so it is equals to minus d divided by a so d is 5 so it becomes minus 5 question we have been asked to find out the cubic equation, cubic equation with roots as 2 alpha, 2 beta, and 2 gamma.

02:23 Okay, so since we know that alpha plus beta plus gamma is equals to 2.

02:30 So if you have to find 2 alpha plus 2 beta plus 2 gamma, we have to multiply with 2 so it becomes 4 year.

02:39 Similarly, alpha, beta, beta, gamma, and gamma alpha, we know it is equal to minus 4.

02:45 Okay.

02:47 So in place of alpha, we have to replace it with 2 alpha and beta with 2 beta.

02:51 So it becomes 4 alpha beta plus 4 beta gamma plus 4 alpha gamma.

02:58 So it will be equals to minus 16...