Determinant of a matrix of polynomials

This question requires that the determinant of this matrix with polynomial entries is equal to 10 for real values of x. The solution employs the use of certain properties of determinants that may not be often talked about.

[devcoolkol]

Listening to the community and making another video is just pure love for what you're doing, thanks.

[wavingbuddy]

for anyone wondering if you want the complex solutions even though they aren't required:

x = (5/6)^(1/3) e^(±(2 i π)/3) and x = e^( ±(i π)/3)

or in trigonometric form:

x = (5/6)^(1/3) (cos((2 π)/3) ± i sin((2 π)/3)) and x = cos(π/3) ± i sin(π/3)

or their cartesian form (which are particularly disgusting):

x = -1/2 (5/6)^(1/3) ± (i 3^(1/6) 5^(1/3))/(2 2^(1/3)) and x = 1/2 ±(i sqrt(3))/2

[nothingbutmathproofs]

This determinant is called the Vandermonde Determinant

[JoachimFavre]

I didn't know those properties of the determinant. Your solution is very elegant, thank you sir! :D

[Harrykesh]

Never thought of doing this way, we are taught(here in India) to use row and column transforms to simplify the matrix or get maximum 0's and then take determinant of both sides

[upalsengupta]

Sir yo are distributing a beautiful thoughts across the globe. Let your knowledge of firey math quench thrust of students of mathematical science. ❤❤❤

[victornassuiro]

Very well demonstrated. Congratulations!

[fadvis]

Realmente maravillosas esas nuevas propiedades del determinante, como aprochando factorizar tanto filas como columnas y luego multiplcando los resultados. Gracias

[kragiharp]

Thank you so much for your explanations!
❤️🙏

[experimentingalgorithm]

Isn't it a standard determinant to be remembered?

[pokemonerpokemoner]

i think he is loving advanced questions.. would love to see him solve more difficult questions of iit jee.. specialy the limits and geometry ones..also the 2024 paper is near..its on 26may 2024

[Bedoroski]

This is amazing to learn. What are these theorems called?

[wychan]

Alternatively, we can just subtract the last row by 3xfirst row and subtract the second row by 2xfirst row, this will make the first element of the second and last row to 0. The determinant then becomes x.|…| where |…| is the lower right hand submatrix.

[PureHanbali]

Shouldn't it be 4, as in there are 3 solutions that are equal to each other & another unique one? Apparently, this is what we are taught in Bangladesh.

[nedeljkosovljanski]

Am I only how saw mistake or there is no mistake? Second matrix should be 0 0 x3 ...