It depends on the purpose of the qn. If, its an objective or a quiz, u can simply put the convenient values of x,y and z.. say 1, -1 and 0. If the question demands solving the expression, then this video is a perfect approach. Hats off!
Wow, an answer is actually not an expression, but a number 1/2. To keep the long story short, the point of this excercise is squaring 3-element expressions: x+y+z, x^2+y^2+z^2 and xy+yz+xz. The fact that x+y+z=0 is also necessary for some simplifications. Actually, if it wasn't 0 but some other number, this whole thing wouldn't simplify that nicely and our final result might not be just a number, but some expression. Well created.
In this video when you find the value of x^4+y^4+z^4=... Then ,if you added and substract x^4+y^4+z^4 you get the answer in next line but still you are good
This is from the Newton-Girard identities for symmetric polynomials. The relevant identities are E1*P1-E0*P2 = 2*E2 E3*P1-E2*P2+E1*P3-E0*P4 = 4*E4 In which E0=1 E1=P1=x+y+z E2=xy+yz+zx E4=0 P2=x^2+y^2+z^2 P4=x^4+y^4+z^4 In this case E1=P1=0 Hence -P2 = 2*E2 -E2*P2-P4 = 0 And P2^2/2 = P4 P4/P2^2 = 1/2
Another example of the power of (elementary) symmetric polynomials. Then you express (p₂)² = (x²+y²+z²)² in two different ways. Very nice! = x²+y²+z² in two different wa
1/2 mentally u just set x=1 y=0 and z=-1 ans u put into the second expression. There is more information implicit in the problem because u know before hand that the result is a number that must hold on for any values of x y and z so u pick the most easy x=1 y=0 z=-1 for example
Initially I thought why ain't we simply assume x=1,y=w,z=w^2...then looked at the expression and used certain limiting values and got the ans. What about it then
@@WillCummingsvideos The limit is certainly defined. The original expression is however not defined at x = y = z = 0. But since the limit is defined, this singularity is removable.
I wonder if there is a simple symmetry argument to see the solution must just be a number without actually doing any work. You sort of mentioned that the answer must be a number but you didn't go into detail why that is, so I am wondering how one would know beforehand.
Nice problem! An aside question... is there some simple way to demonstrate that the expression simplifies to a constant? Then it is justifiable to just substitute an arbitrary (x,y,z) such that x+y+z=0.
@Aletak 13 if it is not must to show the complete process and only answer is required, then according to the given condition you can assume suitable real values of x,y and z put those values in the given expression to get the answer instantly. It will be always right provided u don't assume them all zero.
X+y+z=0 therefore x=z=1, y=-2 is a solution num=x4+y4+z4=1+16+1=16. den=(x2+y2+z2)2= (1+4+1)2=16 hence 16/32=0.5 You don’t need to do any algebra. Just find a solution to x+y=z=0 & it’s easy from there.
It took me 1 minute to come to one solution that gives 1/2 as a result (x=1, y=1, z=-2). Of course, that does not mean there is only 1 single solution.
This whole video is a lie cuz u didnt metioned what would happpen if u choose any two of (x,y,z) 0. Just kidding dude absolutely love your videos please keep urself in business
