Q.5. Factorise:

(i) 4x2 + 9y2 + 16z2 + 12xy - 24yz - 16xz

(ii) 2x2 + y2 + 8z2 - 2√2 xy + 4√2 yz - 8xz

Solution:

(i) Using identity, (a + b + c)= a2 + b2 + c2 + 2ab + 2bc + 2ca

We can write,

4x2 + 9y2 + 16z2 + 12xy - 24yz - 16xz

= (2x)2 + (3y)2 + (-4z)2 + (2×2x×3y) + (2×3y×-4z) + (2×-4z×2x)

= (2x + 3y - 4z)2

= (2x + 3y - 4z) (2x + 3y - 4z)

(ii) Using identity, (a + b + c)= a2 + b2 + c2 + 2ab + 2bc + 2ca

We can write,

2x2 + y2 + 8z2 - 22 xy + 42 yz - 8xz

= (- 2x)2 + (y)2 + (22z)2 + 2×(-2x) ×y + 2×y×22z + 2×22z×(-2x)

= (-2x + y + 22z)2

= (-2x + y + 22z) (-2x + y + 22z)