If a, b, c are roots of x^3-p x+q=0 and Δ=| a(a^2-b c) b(b^2-a c) c(c^2-a b) b

If a, b, c are roots of x^3-p x+q=0 and Δ=| a(a^2-b c) ...

Key Concepts

Symmetric Functions

Symmetric functions are expressions in several variables that remain unchanged under any permutation of the variables. In the context of polynomial roots, these functions allow one to express complicated combinations of roots in terms of elementary symmetric functions, which, by Vieta's formulas, are directly related to the coefficients of the polynomial. This significantly simplifies the manipulation and evaluation of complex algebraic expressions involving roots.

Determinants

Determinants are scalar values that can be computed from a square matrix and encapsulate key properties of the matrix, such as invertibility and the volume scaling factor of linear transformations. They possess various computational properties including expansion by minors, linearity, and column operations which can simplify the evaluation of complex determinants, even when the entries are functions of other variables, such as the roots of a polynomial.

Vieta's Formulas

Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. When the roots are used to form matrix entries or other expressions, these formulas allow one to substitute symmetric sums and products, thereby simplifying the expressions by reducing them to functions of the polynomial’s coefficients. This approach is particularly useful in problems involving roots of polynomials where direct computation would be cumbersome.

Transcript

00:01 So in the given question we are told that a, b and c, a, b and c are roots of the cubic equation that is given as x cube minus px plus q as equal to 0.

00:25 Right? so one thing that we can write for a cubic equation is we can equate the coefficient that is given for x square, that is we can write this cubic equation as x cube plus zero x squared minus p x plus q equal to zero and since a b c are roots of this equation a property of the of the roots is that we can write a.

00:57 Plus b plus c as the negative coefficient of x squared over here it is equal to 0 so we can write a plus b plus c would be equal to 0 so this is a property of a cubic of the roots of a cubic equation right so now in the question we are given a determinant that is given as a times a square minus b c b times b squared minus a c b times b squared minus a c c times c squared minus a b and in the second row we have b times b plus 2c c times c plus 2a a times a plus 2b in the third row we have 3c minus 2a 3a minus 2b and in the third row we have 3 b minus 2b and in the third row we have 3b minus 2c.

02:03 So this is what we have as the determinant right and we are told to simplify this determinant if a b c are given as the roots of the given cubic equation right so what we can do over here is to first do a row operation which is a row operation which we can write as r1 changes to r1 changes to r1 plus r2 plus r3.

02:48 R1 plus r2 plus r3, right? so when we do this, what we would get is we would have a cube minus a b b c plus b square plus 2 bc plus 3 c plus 3 c.

03:08 Minus 2a in over here we would have b cube minus a bc plus we have plus c square plus 2 a c plus 3a minus 2b and the third element is c cube minus abc plus a square plus 2 a b plus 3b minus 2c right so we have this and the rest of the rest of the determinant is the same right the rest of the determinant is the same so from this what we are going to do is the in the next the operation what we are going to do is the in the next operation what we are going to to do is you are going to take this c1 column c1 and add c1 plus c2 plus c3 right and once we do that we will have in column 1 let's fill out the determinant first so we have b plus me plus d times b plus 2a c times c plus 2a and a times c plus 2a and a times a plus 2b over here and the third row is also unchanged with this which is 3c minus 2a 3a minus 2b and 3b plus minus 2c right so when we do the column operation c1 plus c2 plus c3 what we will have is a cube plus b cube plus c cube minus 3 a b bc cube minus 3 a b bc plus a square plus b square plus a square plus a square plus b square plus c square plus c square plus two bc plus two bc plus two ac and we will have plus two ab plus two ab what is this a cube plus 2 plus c cube minus 3 a b c right so a square plus b square plus c plus c plus b square plus c plus b c plus b plus 2 a b and we have next what we have is 3c minus 2c is c 3a minus 2a is a so we have a plus b plus c right we can write plus a plus a plus a plus b plus c so this is the first element in the first column and the second element is b square plus 2 a b plus c square plus 2 a c plus a square plus 2 a b so we can rewrite it as a square plus b square plus c square plus 2ab plus 2 ab plus 2 bc 2ab plus 2 ac 2a c plus 2ab.

07:10 Right? so this is what we have.

07:15 And next in the third element in the third column is over here it is 2c, right? it is 2c over here, 2c.

07:33 So we will have 2bc here to b c and in the third element in the first column is 3c plus minus 2c is c so we will have a plus b plus c as the third element in the third row right so what we can do over here is in this step we can take a plus b plus c as a common factor from the first column over here so let's write the rest of the determinant which is actually the same right so we can take this part which is actually the same we can take this this has no change we have we have just done the column operation on the first column so we have no change in the first in the second and third column right so let's select this this part over here we can select this and we can just keep this over here so now we have a complete determinant right so this is the determinant that we have and from this since we did two operations that is first we did the row operation and then we did the column operation.

09:23 Now what we can see that the first element is actually the expansion of a plus b plus c whole cube...

Please give Ace some feedback