Question

Use the binomial theorem to find the expansion of ( x - y + z)^3

Answer 1

The expansion of (x - y + z)^3 using the binomial theorem is:

(x - y + z)^3 = C(3,0)x^3 + C(3,1)x^2(-y) + C(3,2)x(-y)^2 + C(3,3)(-y)^3 + C(3,0)x^2z + C(3,1)x(-y)z + C(3,2)(-y)^2z + C(3,0)xz^2 + C(3,1)(-y)z^2 + C(3,0)z^3

where C(n,k) is the binomial coefficient, given by:

C(n,k) = n! / (k! * (n-k)!)

Substituting the values, we get:

(x - y + z)^3 = 1x^3 - 3x^2y + 3xy^2 - 1y^3 + 3x^2z - 6xyz + 3y^2z + 3xz^2 - 3yz^2 + 1z^3

Therefore, the expansion of (x - y + z)^3 using the binomial theorem is:
x^3 - 3x^2y + 3xy^2 - y^3 + 3x^2z - 6xyz + 3y^2z + 3xz^2 - 3yz^2 + z^3.