Which is the correct factorisation of x³ + y³ + z³ − 3xyz?

Which is the correct factorisation of $x^{3} + y^{3} + z^{3} - 3 x yz$ ?

Class 9PolynomialsPolynomials

x^{3} + y^{3} + z^{3} - 3 x yz = (x + y + z) (x^{2} + y^{2} + z^{2} - x y - yz - z x)

. All three product terms in the second bracket carry a minus sign. This is also why

x^{3} + y^{3} + z^{3} = 3 x yz

whenever

x + y + z = 0

— the first bracket becomes 0.

Answer: C)

(x + y + z) (x^{2} + y^{2} + z^{2} - x y - yz - z x)

**Linear**: ax +

b = 0

→

x = - b / a

. One zero.

Example:

3 x - 12 = 0

→

x = 4

.

**Quadratic**: solve by factorisation, or quadratic formula (Class 10).

Example:

x^{2} - 5 x + 6 = 0

. Factor:

(x - 2) (x - 3) = 0

→

x = 2

x = 3

. Two zeros.

**General principle**: a polynomial of degree n has AT MOST n real zeros.

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