If α, β, Yare the roets of 2 x^3-3 x^2+6 x+1=0, then u^2+β^2+γ^2 is (a) 15 / 4 (b) -3 (c) -15 /

If α, β, Yare the roets of 2 x^3-3 x^2+6 x+1=0, then...

Key Concepts

Vieta's Formulas

Vieta's formulas relate the roots of a polynomial to its coefficients. For a cubic polynomial, the formulas express the sum of the roots, the sum of the products of the roots taken two at a time, and the product of the roots in terms of the coefficients. This relationship allows one to determine combinations of the roots without solving for the roots explicitly.

Sum of Squares of Roots

The sum of the squares of the roots of a polynomial can be calculated using an identity that involves the sum of the roots and the sum of the products of the roots taken two at a time. Specifically, for roots a, b, and c, the identity a² + b² + c² = (a + b + c)² - 2(ab + ac + bc) is used. This approach utilizes Vieta's formulas to express these sums solely in terms of the polynomial's coefficients.

Transcript

00:01 In this video, we will try and find the values of the alpha squared plus beta squared plus gamma squared if alpha, beta, gamma are the roots of the following equation.

00:11 As you can see, it's a cubic equation, therefore there are three roots in the form of alpha, beta, and gamma.

00:17 Before we try and find out the values of these roots, it's important to realize that for a cubic equation, there are important identities that can be established in the form of the roots of the equation, as against the coefficient.

00:31 Of these equations.

00:33 Okay.

00:35 So let's look at some of the essential formula that would be important for a cubic equation.

00:41 So for instance, for a generic cubic equation of this form, a x -q plus b squared plus c x plus d is equal to zero.

00:50 Assuming that the roots are in the form of alpha beat d and gamma, it's possible to establish the three relationship in the form of the sum of the roots, the product, pair, of the roots and product of the roots.

01:04 So they would satisfy exactly these conditions.

01:08 That is the sum of the roots in this case there are three roots.

01:11 Alpha beta, alpha, alpha plus beta plus gamma would translate into minus the coefficient of x squared by coefficient of x cube which is in the form of minus b by a.

01:20 Some of the product pairs of the roots in the form of alpha beta plus beta plus beta plus gamma alpha will translate into the coefficient of x divided by the coefficient of x cube which is c by a.

01:32 The product of the roots, which is alpha, beta, gamma, that is the multiplication of alpha, multiplied by beta, multiple gamma, will translate to the minus of the constant, which is in this case it's d, the constant is d, divided by the coefficient of x cube, which would be a, which is minus d by a...

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