(i) $x^{2} + y^{2} + z^{2} - x y + x + y z by x + y - z$

(ii) $x^{2} + 4 y^{2} + z^{2} + 2 x y + x z - 2 y z by x - 2 y - z$

(iii) $x^{2} + 4 y^{2} + 2 x y - 3 x + 6 y + 9 by x - 2 y + 3$

(iv) $9 x^{2} + 25 y^{2} + 15 x y + 2 x - 20 y + 16 by 3 x - 5 y + 4$

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Solution

$(i) (x^{2} + y^{2} + z^{2} - x y + y z + z x) b y (x + y - z) - x^{3} + y^{3} - z^{3} + 3 x y z (i i) (x^{2} + 4 y^{2} + z^{2} + 2 x y + x z - 2 y z) b y (x - 2 y - z) = (x - 2 y - z) [x^{2} + (- 2 y)^{2} + (- z)^{2} - x \times (- 2 y) - (- 2 y) (z) - (- z) (x)] = x^{3} + (- 2 y)^{3} + (- z)^{3} - 3 x (- 2 y) (- z) = x^{3} - 8 y^{3} - z^{3} - 6 x y z (i i i) x^{2} + 4 y^{2} + 2 x y - 3 x + 6 y + 9 b y x - 2 y + 3 = (x - 2 y + 3) (x^{2} + 4 y^{2} + 9 + 2 x y + 5 y - 3 x) = (x)^{3} + (- 2 y)^{2} + (3)^{3} - 3 \times x \times (- 2 y) \times 3 = x^{3} - 8 y^{3} + 27 + 18 x y (i v) 9 x^{2} + 14 y^{2} + 15 x y + 12 x - 20 y + 16 b y 3 x - 5 y + 4 = (3 x - 5 y + 4) [(3 x)^{2} + (- 5 y)^{2} + (4)^{2} - 3 x \times (- 5 y) (- 5 y \times 4) - (4 \times 3 x)] = (3 x)^{3} + (- 5 y)^{3} + (4)^{3} - 3 \times 3 x (- 5 y) \times 4 = 27 x^{3} - 125 y^{3} + 64 + 180 x y$

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