If p, q, r are distinct, p+q+r=0 and

Key Concepts

Cubic Sum Factorization

The factorization identity for cubes, namely a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca), is essential for understanding how sums of cubes relate to the product of the variables. This identity is especially useful in problems where the sum a + b + c is known to assume a specific value, such as zero, leading to significant simplifications.

Symmetric Polynomials

Symmetric polynomials are expressions in several variables that remain invariant under any permutation of the variables. This invariance makes them very useful in algebra, particularly in solving equations and finding relationships between the roots of polynomials, as well as in factorization.

Zero Determinant Condition

When a matrix has a determinant equal to zero, it implies that the vectors corresponding to the matrix's rows or columns are linearly dependent. This relationship imposes specific constraints on the variables involved and often leads to conditions that simplify the relationships between these variables in algebraic problems.

Determinant of a Matrix

A determinant is a scalar value derived from a square matrix that encodes key properties of the linear transformation represented by the matrix, such as scaling factors and orientation. It is widely used to determine whether a matrix is invertible. A zero determinant indicates that the matrix is singular, reflecting linear dependence among its rows or columns.

Transcript

00:01 So in the given question we are told that if the numbers p, q and r are distinct numbers are distinct numbers and it is given that p plus q plus r is equal to 0 and the determinant of pa, pa, qb, r c, qc ra, tb, b b and it is a tb and it is a, and it is a, and the third row we have r b b c and q a this determinant is equal to zero then we are told to find the value of a cube plus b cube plus c cube right so what we can do over here is to first evaluate this determinant right so when we evaluate this determinant we would have of pa times qr a square minus b square b c minus qb times q square ac minus p square b square pr, b square pr plus we have rc times c square pq minus r square ab minus r square ab minus r square ab and this is equal to 0 so when we expand this we would have pq r a cube minus abc bc bcccccccccc cube plus pqr bccccccccccc plus p qrcccccccc minus p q a, b, c, r, cube equal to 0.

02:09 We can take pqr and abc as common terms.

02:14 We can take pqr as a common term from these terms, right? from the terms, these terms.

02:28 And then we would have pqr times a2 plus pq plus c cube.

02:32 And we can take minus a bc as a common factor from these terms.

02:37 Terms and we would have minus a b c times bc plus q q plus r cube and this is equal to zero so now we can use an identity that says if we have a plus b plus c sorry a cube plus b2 plus c can be found as can be found as a plus b plus c times a square plus b square plus c square minus a b square minus a c minus b c plus three times a b c so using this identity we can expand p q plus q plus q plus r q right so when we do that what we would get is p q plus q q plus r cube is equal to p plus q plus r times p square plus q square plus r minus p q minus qr minus p r minus p r plus three times p qr so this is what we would have but in the question we have we are already told that p plus q plus r has the value zero, right? then this whole term in this expansion would reduce to zero, which means we would have pq plus q plus r cube plus r cube is equal to three times pqr.

04:32 So now we can substitute that in the above equation and we would have pqr.

04:42 Times a cube plus b cube plus c cube minus a bc times three times pqr right and this is equal to zero so again we can take pqr as a common factor from this and when we take pqr as a common factor we would have pqr times a cube plus b cubed a cube plus b cube plus b cube plus c cube minus 3 a bc is equal to 0 3 a bc is equal to 0 so p q r or p q and r are distinct numbers which means p times q times r cannot be equal to 0 since p is not equal to 0, p is not equal to q not equal to r, p times q times r cannot be equal to 0, right? so from the conditions that are given in the question that pq and r are different numbers and p plus q plus r is equal to 0, it can't be that when we take the product of pq and r we would get 0...

Please give Ace some feedback