Question

if X + Y + Z is equal to 5 and xy + y z + by X is equals to 10 then prove that x cube + Z cube minus 3 x y z is equal to -25​

Answer 1

Question (with correction)

If x+y+z =5 and xy+yz+zx =10 then prove that x³+y³+z³-3xyz = -25

Given:

x+y+z =5

xy+yz+zx =10

To prove:

x³+y³+z³-3xyz = -25

Solution:

We know that

(x+y+z)² = x²+y²+z²+2(xy+yz+zx)

(5)² = x²+y²+z²+2(10)

25= x²+y²+z²+20

5 = x²+y²+z²

Now we have the value of x² +y² +z²= 5

Again using identity

x³+y³+z³-3xyz = (x+y+z)(x²+y²+z²-xy-yz-zx)

we will substitute the values

x³+y³+z³-3xyz = [ (x+y+z)(x²+y²+z²- (xy+yz+zx) ]

x³+y³+z³-3xyz = [ (5)(x²+y²+z²) -(10) ]

x³+y³+z³-3xyz = 5(5-10)

x³+y³+z³-3xyz = 5×(-5)

x³+y³+z³-3xyz = -25

=> x³+y³+z³ -3xyz = -25

Hence proved