Polynomials — Question 5

(i) 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 [∵ (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx]

Hence, 4x² + 9y² + 16z² + 12xy - 24yz - 16xz = (2x + 3y - 4z)(2x + 3y - 4z)

(ii) 2x² + y² + 8z² - $2\sqrt{2}$xy + $4\sqrt{2}$yz - 8xz

[∵ (a + b + c)² = (a)² + (b)² + (c)² + 2ab + 2bc + 2ca]

$= (-\sqrt{2}x)^2 + (y)^2 + (2\sqrt{2}z)^2 + 2(-\sqrt{2}x)(y) + 2(y)(2\sqrt{2}z) + 2(2\sqrt{2}z)(-\sqrt{2}x) \\[1em] = (-\sqrt{2}x + y + 2\sqrt{2}z)^2$

Hence, 2x² + y² + 8z² -$2\sqrt{2}$xy + $4\sqrt{2}$yz - 8zx = (-$\sqrt{2}$x + y + $2\sqrt{2}$z)(-$\sqrt{2}$x + y + $2\sqrt{2}$z)